Infinity is that which is boundless, endless, or larger than any

_{0}), the cardinality of the set of

Ancient Jaina Mathematics: an Introduction

'

Infinity Foundation

* Bell, J.L.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 2009. *. * * Jain, L.C. (1973). "Set theory in the Jaina school of mathematics", ''Indian Journal of History of Science''. * * H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html * * O'Connor, John J. and Edmund F. Robertson (1998)

, ''MacTutor History of Mathematics archive''. * O'Connor, John J. and Edmund F. Robertson (2000)

'Jaina mathematics'

, ''MacTutor History of Mathematics archive''. * Pearce, Ian. (2002)

''MacTutor History of Mathematics archive''. * *

'', 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. *

'', 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. *

* John J. O'Connor and Edmund F. Robertson (1998)

, ''MacTutor History of Mathematics archive''. * John J. O'Connor and Edmund F. Robertson (2000)

'Jaina mathematics'

, ''MacTutor History of Mathematics archive''. * Ian Pearce (2002)

''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]

Dictionary of the Infinite

(compilation of articles about infinity in physics, mathematics, and philosophy) {{Authority control Infinity, Concepts in logic Philosophy of mathematics Mathematical objects

number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

. It is often denoted by the infinity symbol
The infinity symbol (\infty, , or ∞) is a mathematical symbol representing the concept of infinity. In algebraic geometry, the figure is called a lemniscate.
History
The shape of a sideways figure eight has a long pedigree; for instance, it ...

.
Since the time of the ancient Greeks
Ancient Greece ( el, Ἑλλάς, Hellás) was a civilization belonging to a period of History of Greece, Greek history from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, antiquity ( AD 600). This era was ...

, the philosophical nature of infinity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

, mathematicians began to work with infinite series
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

and what some mathematicians (including and BernoulliBernoulli can refer to:
People
*Bernoulli family of 17th and 18th century Swiss mathematicians:
** Daniel Bernoulli (1700–1782), developer of Bernoulli's principle
** Jacob Bernoulli (1654–1705), also known as Jacques, after whom Bernoulli numbe ...

) 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 and, if so, how this could be done. At the end of the 19th century, Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...

enlarged the mathematical study of infinity by studying infinite set
In set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...

s and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of the line) is larger than the number of integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...

s. In this usage, infinity is a mathematical concept, and infinite mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs ...

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
In set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any t ...

, on which most of modern mathematics can be developed, is the axiom of infinity
In axiomatic set theory and the branches of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...

, which guarantees the existence of infinite sets. The mathematical concept of infinity and the manipulation of infinite sets are used everywhere in mathematics, even in areas such as combinatorics
Combinatorics is an area of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...

that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than 2. The cases ...

implicitly relies on the existence of very large infinite sets for solving a long-standing problem that is stated in terms of elementary arithmetic
Elementary arithmetic is the simplified portion of arithmetic that includes the operations of addition, subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is sig ...

.
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

and cosmology
Cosmology (from Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

, whether the Universe is infinite is an open question.
History

Ancient cultures had various ideas about the nature of infinity. The ancient Indians andGreeks
The Greeks or Hellenes (; el, Έλληνες, ''Éllines'' ) are an ethnic group
An ethnic group or ethnicity is a grouping of people
A people is any plurality of person
A person (plural people or persons) is a being that has cer ...

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 may be that ofAnaximander
Anaximander (; grc-gre, Ἀναξίμανδρος ''Anaximandros''; ) was a who lived in ,"Anaximander" in '. London: , 1961, Vol. 1, p. 403. a city of (in modern-day Turkey). He belonged to the and learned the teachings of his master . He s ...

(c. 610 – c. 546 BC) a pre-Socratic
Pre-Socratic philosophy is ancient Greek philosophy
Ancient Greek philosophy arose in the 6th century BC, at a time when the inhabitants of ancient Greece were struggling to repel devastating invasions from the east. Greek philosophy continued t ...

Greek philosopher. He used the word ''apeiron'', which means "unbounded", "indefinite", and perhaps can be translated as "infinite".
Aristotle (350 BC) distinguished ''potential infinity'' from ''actual infinity
In the philosophy of mathematics
Philosophy (from , ) is the study of general and fundamental questions, such as those about reason
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient ...

'', which he regarded as impossible due to the various paradoxes it seemed to produce. It has been argued that, in line with this view, the Hellenistic
The Hellenistic period spans the period of Mediterranean history
The Mediterranean Sea is a sea connected to the Atlantic Ocean, surrounded by the Mediterranean Basin and almost completely enclosed by land: on the north by Western Europe, We ...

Greeks had a "horror of the infinite" which would, for example, explain why Euclid
Euclid (; grc-gre, Εὐκλείδης
Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

(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." It has also been maintained, that, in proving the infinitude of the prime numbers
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime
A prime number (or a prime) is a natural number
In mathematics, the natural numbers are those used for counting (as in "there ...

, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's parallel postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:
' ...

, sometimes translated
:If a straight line falling across two straight lines makes internal angles on the same side f itself whose sum isless than two right angles, then the two straight lines, being produced to infinity, meet on that side f the original straight linethat the um of the internal anglesis less than two right angles.
Other translators, however, prefer the translation "the two straight lines, if produced indefinitely ...", 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.
Zeno: Achilles and the tortoise

( 495 – 430 BC) did not advance any views concerning the infinite. Nevertheless, his paradoxes, especially "Achilles and the Tortoise", were important contributions in that they made clear the inadequacy of popular conceptions. The paradoxes were described byBertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British , , , , , , , , and .Stanford Encyclopedia of Philosophy"Bertrand Russell" 1 May 2003 Throughout his life, Russell considered himself a , a and ...

as "immeasurably subtle and profound".
Achilles
In Greek mythology
Greek mythology is the body of s originally told by the , and a of . These stories concern the and , the lives and activities of , , and , and the origins and significance of the ancient Greeks' own and practices. ...

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
The Eleatics were a Pre-Socratic philosophy, pre-Socratic school of philosophy founded by Parmenides in the early fifth century BC in the ancient town of Velia, Elea. Other members of the school included Zeno of Elea and Melissus of Samos. Xen ...

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
Baron
Baron is a rank of nobility or title of honour, often hereditary, in various European countries, either current or historical. The female equivalent is baroness. Typically, the title denotes an aristocrat who ranks higher than a lord ...

provided both a satisfactory definition of a limit and a proof that, for ,
:
Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meter per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with and . Achilles does overtake the tortoise; it takes him
:$10+0.1+0.001+0.00001+\backslash cdots=\backslash frac\; =\; \backslash frac\; =10.10101\backslash ldots\backslash text.$
Early Indian

The 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: * Enumerable: lowest, intermediate, and highest * Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable * Infinite: nearly infinite, truly infinite, infinitely infinite17th century

In the 17th century, European mathematicians started using infinite numbers and infinite expressions in a systematic fashion. In 1655,John Wallis
John Wallis (; la, Wallisius; ) was an English clergyman and mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), f ...

first used the notation $\backslash infty$ for such a number in his ''De sectionibus conicis,'' and exploited it in area calculations by dividing the region into infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...

strips of width on the order of $\backslash tfrac.$ But in ''Arithmetica infinitorum'' (also in 1655), 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."
In 1699, Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics a ...

wrote about equations with an infinite number of terms in his work ''De analysi per aequationes numero terminorum infinitas
''De analysi per aequationes numero terminorum infinitas'' (or ''On analysis by infinite series'', ''On Analysis by Equations with an infinite number of terms'', or ''On the Analysis by means of equations of an infinite number of terms'', is a mat ...

''.
Mathematics

Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German
German(s) may refer to:
Common uses
* of or related to Germany
* Germans, Germanic ethnic group, citizens of Germany or people of German ancestry
* For citizens of ...

opened a mathematico-philosophic address given in 1930 with:
Symbol

The infinity symbol $\backslash infty$ (sometimes called thelemniscate
400px, The lemniscate of Bernoulli and its two foci
In algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the us ...

) is a mathematical symbol representing the concept of infinity. The symbol is encoded in Unicode
Unicode, formally the Unicode Standard, is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expressed in most of the world's wri ...

at and in LaTeX
Latex is a stable dispersion (emulsion
An emulsion is a mixture of two or more liquids that are normally Miscibility, immiscible (unmixable or unblendable) owing to liquid-liquid phase separation. Emulsions are part of a more general class o ...

as `\infty`

.
It was introduced in 1655 by John Wallis
John Wallis (; la, Wallisius; ) was an English clergyman and mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), f ...

, and since its introduction, it has also been used outside mathematics in modern mysticism and literary symbology
A symbol is a mark, sign, or word
In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical meaning (linguistics), m ...

.
Calculus

Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist, and diplomat. He is a promin ...

, one of the co-inventors of infinitesimal calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

, 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
The law of continuity is a heuristic principle introduced by Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. (; or ; – 14 November 1716) was a prominent German p ...

.
Real analysis

Inreal analysis
200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.">square_wave.html" ;"title="Fourier series for a square wave">Fourier series for a square wave. Fourier series are a ...

, the symbol $\backslash infty$, called "infinity", is used to denote an unbounded limit
Limit or Limits may refer to:
Arts and media
* Limit (music), a way to characterize harmony
* Limit (song), "Limit" (song), a 2016 single by Luna Sea
* Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...

. The notation $x\; \backslash rightarrow\; \backslash infty$ means that ''$x$'' increases without bound, and $x\; \backslash to\; -\backslash infty$ means that ''$x$'' decreases without bound. For example, if $f(t)\backslash ge\; 0$ for every ''$t$'', then
* $\backslash int\_^\; f(t)\backslash ,\; dt\; =\; \backslash infty$ means that $f(t)$ does not bound a finite area from $a$ to $b.$
* $\backslash int\_^\; f(t)\backslash ,\; dt\; =\; \backslash infty$ means that the area under $f(t)$ is infinite.
* $\backslash int\_^\; f(t)\backslash ,\; dt\; =\; a$ means that the total area under $f(t)$ is finite, and is equal to $a.$
Infinity can also be used to describe infinite series
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, as follows:
* $\backslash sum\_^\; f(i)\; =\; a$ means that the sum of the infinite series converges to some real value $a.$
* $\backslash sum\_^\; f(i)\; =\; \backslash infty$ means that the sum of the infinite series properly diverges to infinity, in the sense that the partial sums increase without bound.
In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled $+\backslash infty$ and $-\backslash infty$ can be added to the topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s. We can also treat $+\backslash infty$ and $-\backslash infty$ as the same, leading to the one-point compactificationIn the mathematical field of topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematic ...

of the real numbers, which is the real projective lineImage:Real projective line.svg, The real projective line can be modeled by the projectively extended real line, which consists of the real line together with a point at infinity; i.e., the one-point compactification of R.
In geometry, a real projecti ...

. Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, proj ...

also refers to a line at infinity
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

in plane geometry, a plane at infinity
In projective geometry, a plane at infinity is the hyperplane at infinity of a three dimensional projective space or to any Plane (geometry), plane contained in the hyperplane at infinity of any projective space of higher dimension. This article wil ...

in three-dimensional space, and a hyperplane at infinityIn geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

for general dimensions
thumb
, 236px
, The first four spatial dimensions, represented in a two-dimensional picture.
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature ...

, each consisting of points at infinity
150px, The real line with the point at infinity; it is called the real projective line.
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic ...

.
Complex analysis

Incomplex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis
Analysis is the branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such ...

the symbol $\backslash infty$, called "infinity", denotes an unsigned infinite limit
Limit or Limits may refer to:
Arts and media
* Limit (music), a way to characterize harmony
* Limit (song), "Limit" (song), a 2016 single by Luna Sea
* Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...

. $x\; \backslash rightarrow\; \backslash infty$ means that the magnitude $,\; x,$ of ''$x$'' grows beyond any assigned value. A point labeled $\backslash infty$ can be added to the complex plane as a topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

giving the one-point compactificationIn the mathematical field of topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematic ...

of the complex plane. When this is done, the resulting space is a one-dimensional complex manifold
In differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential ...

, or Riemann surface
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

, called the extended complex plane or the Riemann sphere
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. 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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

, namely $z/0\; =\; \backslash infty$ for any nonzero complex number ''$z$''. In this context, it is often useful to consider meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function (mathematics), function that is holomorphic function, holomorphic on all of ''D'' ''except'' for a set of isolated p ...

s as maps into the Riemann sphere taking the value of $\backslash infty$ at the poles. The domain of a complex-valued function may be extended to include the point at infinity as well. One important example of such functions is the group of Möbius transformation
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

s (see ).
Nonstandard analysis

The original formulation ofinfinitesimal calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

by Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics a ...

and Gottfried Leibniz used infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...

quantities. In the 20th century, it was shown that this treatment could be put on a rigorous footing through various logical system
A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A formal system is essentiall ...

s, including smooth infinitesimal analysisSmooth infinitesimal analysis is a modern reformulation of the calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer ...

and nonstandard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...

. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian 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 .
Set theory

A different form of "infinity" are theordinal
Ordinal may refer to:
* Ordinal data, a statistical data type consisting of numerical scores that exist on an arbitrary numerical scale
* Ordinal date, a simple form of expressing a date using only the year and the day number within that year
* Or ...

and cardinal
Cardinal or The Cardinal may refer to:
Christianity
* Cardinal (Catholic Church), a senior official of the Catholic Church
* Cardinal (Church of England), two members of the College of Minor Canons of St. Paul's Cathedral
Navigation
* Cardina ...

infinities of set theory—a system of transfinite number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s first developed by Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence be ...

. In this system, the first transfinite cardinal is aleph-null
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

(ℵnatural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s. This modern mathematical conception of the quantitative infinite developed in the late 19th century from works by Cantor, Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He worked as a mathematics professor at the University of Jena, and is understood by many to be the father of analy ...

, Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory
In algebra, ring theory is the study of ring (mathematics), rings ...

and others—using the idea of collections or sets.
Dedekind's approach was essentially to adopt the idea of one-to-one correspondence
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

as a standard for comparing the size of sets, and to reject the view of Galileo (derived from Euclid
Euclid (; grc-gre, Εὐκλείδης
Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

) that the whole cannot be the same size as the part (however, see Galileo's paradox where he concludes that positive square integers are of the same size as positive integers). An infinite set can simply be defined as one having the same size as at least one of its proper parts; this notion of infinity is called Dedekind infinite. 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".
Cantor defined two kinds of infinite numbers: ordinal number
In set theory
Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ...

s and cardinal number
150px, Aleph null, the smallest infinite cardinal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...

s. Ordinal numbers characterize well-ordered
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s which are maps from the positive integers
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...

leads to 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 countably infinite
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called ''uncountable
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

''. Cantor's views prevailed and modern mathematics accepts actual infinity as part of a consistent and coherent theory. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
Cardinality of the continuum

One of Cantor's most important results was that the cardinality of the continuum $\backslash mathbf\; c$ is greater than that of the natural numbers $$; that is, there are more real numbers than natural numbers . Namely, Cantor showed that $\backslash mathbf=2^>$. Thecontinuum hypothesis
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

states that there is no cardinal number
150px, Aleph null, the smallest infinite cardinal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...

between the cardinality of the reals and the cardinality of the natural numbers, that is, $\backslash mathbf=\backslash aleph\_1=\backslash beth\_1$.This hypothesis cannot be proved or disproved within the widely accepted Zermelo–Fraenkel set theory
In set theory
illustrating the intersection (set theory), intersection of two set (mathematics), sets.
Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any t ...

, even assuming the Axiom of Choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

.
Cardinal arithmetic
150px, Aleph null, the smallest infinite cardinal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...

can be used to show not only that the number of points in a real number line
Real may refer to:
Currencies
* Brazilian real
The Brazilian real ( pt, real, plural, pl. '; currency symbol, sign: R$; ISO 4217, code: BRL) is the official currency of Brazil. It is subdivided into 100 centavos. The Central Bank of Brazil i ...

is equal to the number of points in any , but also that this is equal to the number of points on a plane and, indeed, in any finite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e. the number of vectors) of a Basis (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after ...

space.
The first of these results is apparent by considering, for instance, the tangent
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

function, which provides a one-to-one correspondence
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

between the interval () and.The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when Giuseppe Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

introduced the space-filling curve
In mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathemat ...

s, curved lines that twist and turn enough to fill the whole of any square, or cube
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

, or hypercube
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

, 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.
Geometry

Until the end of the 19th century, infinity was rarely discussed ingeometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

, except in the context of processes that could be continued without any limit. For example, a line
Line, lines, The Line, or LINE may refer to:
Arts, entertainment, and media Films
* ''Lines'' (film), a 2016 Greek film
* ''The Line'' (2017 film)
* ''The Line'' (2009 film)
* ''The Line'', a 2009 independent film by Nancy Schwartzman
Lite ...

was what is now called a line segment
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

, 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
Locus (plural loci) is Latin for "place". It may refer to:
Entertainment
* Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front
* Locus (magazine), ''Locus'' (magazine), science fiction and fantasy magazine
...

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
In the philosophy of mathematics
Philosophy (from , ) is the study of general and fundamental questions, such as those about reason
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient ...

was projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, proj ...

, where points at infinity
150px, The real line with the point at infinity; it is called the real projective line.
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic ...

are added to the Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...

for modeling the perspective effect that shows parallel lines
In geometry, parallel lines are line (geometry), lines in a plane (geometry), plane which do not meet; that is, two straight lines in a plane that do not intersecting lines, intersect at any point are said to be parallel. Colloquially, curves tha ...

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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, two distinct lines
Long interspersed nuclear elements (LINEs) (also known as long interspersed nucleotide elements or long interspersed elements) are a group of non-LTR (long terminal repeat
A long terminal repeat (LTR) is a pair of identical sequences of DNA
...

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
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...

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 spaces that occur in classicalgeometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

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 spaces are generally vector spaces of infinite dimension.
In topology, some constructions can generate topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

s of infinite dimension. In particular, this is the case of iterated loop spaces.
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.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.Physics

Inphysics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

, approximations of real numbers are used for Continuum (theory), continuous measurements and natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

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.
Cosmology

The first published proposal that the universe is infinite came from Thomas Digges in 1576. 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." 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. 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. 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. 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.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."Computing

The IEEE 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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

, and other exceptional operations.
Some programming languages, such as Java (programming language), Java and J (programming language), J, 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 values in algorithms involving sorting, Search algorithm, searching, or window function, windowing.
In languages that do not have greatest and least elements, but do allow operator overloading, overloading of relational operators, 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.
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 utilizes the concept of vanishing points, 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. Artist M.C. Escher is specifically known for employing the concept of infinity in his work in this and other ways. Variations of chess played on an unbounded board are called infinite chess. 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 (BMI), defined as the ever-increasing sequence <1,2,3,...>. The symbol is often used romantically to represent eternal love. Several types of jewelry are fashioned into the infinity shape for this purpose.See also

* 0.999... * Aleph number * Ananta (infinite), Ananta * Exponentiation * Indeterminate form * Infinite monkey theorem * Infinite set * Infinitesimal * Paradoxes of infinity * Supertask * Surreal numberReferences

Bibliography

* * * * * * * * * *Sources

* *D.P. Agrawal (2000).Ancient Jaina Mathematics: an Introduction

'

Infinity Foundation

* Bell, J.L.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 2009. *. * * Jain, L.C. (1973). "Set theory in the Jaina school of mathematics", ''Indian Journal of History of Science''. * * H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html * * O'Connor, John J. and Edmund F. Robertson (1998)

, ''MacTutor History of Mathematics archive''. * O'Connor, John J. and Edmund F. Robertson (2000)

'Jaina mathematics'

, ''MacTutor History of Mathematics archive''. * Pearce, Ian. (2002)

''MacTutor History of Mathematics archive''. * *

External links

* * *'', 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. *

'', 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. *

* John J. O'Connor and Edmund F. Robertson (1998)

, ''MacTutor History of Mathematics archive''. * John J. O'Connor and Edmund F. Robertson (2000)

'Jaina mathematics'

, ''MacTutor History of Mathematics archive''. * Ian Pearce (2002)

''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]

Dictionary of the Infinite

(compilation of articles about infinity in physics, mathematics, and philosophy) {{Authority control Infinity, Concepts in logic Philosophy of mathematics Mathematical objects