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

number
. 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 ...

infinity symbol
. 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
l'Hôpital
l'Hôpital
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 ...

Bernoulli
) 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 ...

physics
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 and
Greeks 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 of
Anaximander 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 ...

Anaximander
(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 ...

Hellenistic
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 ...

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." 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: ' ...

parallel postulate
, 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 by
Bertrand 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. ...

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 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 ...

Augustin-Louis Cauchy
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+\cdots=\frac = \frac =10.10101\ldots\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 infinite


17th 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 ...

John Wallis
first used the notation \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 \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 ...

Isaac Newton
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 ...

Hermann Weyl
opened a mathematico-philosophic address given in 1930 with:


Symbol

The infinity symbol \infty (sometimes called the
lemniscate 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 ...

lemniscate
) 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 ...

Unicode
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 ...

LaTeX
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 ...

John Wallis
, 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 ...

Gottfried Leibniz
, 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

In
real 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 ...

real analysis
, the symbol \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 ...

limit
. The notation x \rightarrow \infty means that ''x'' increases without bound, and x \to -\infty means that ''x'' decreases without bound. For example, if f(t)\ge 0 for every ''t'', then * \int_^ f(t)\, dt = \infty means that f(t) does not bound a finite area from a to b. * \int_^ f(t)\, dt = \infty means that the area under f(t) is infinite. * \int_^ f(t)\, 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: * \sum_^ f(i) = a means that the sum of the infinite series converges to some real value a. * \sum_^ f(i) = \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 +\infty and -\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 +\infty and -\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 ...

real projective line
.
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 ...

line at infinity
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

In
complex 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 \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 \rightarrow \infty means that the magnitude , x, of ''x'' grows beyond any assigned value. A point labeled \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 ...

Riemann sphere
. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (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 = \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 \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 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 ...
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 ...

Isaac Newton
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 the
ordinal 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 ...

aleph-null
(0), the cardinality of the set of
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. 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 ...

Euclid
) 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 ...

sequence
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 ...

integers
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 \mathbf c is greater than that of the natural numbers ; that is, there are more real numbers than natural numbers . Namely, Cantor showed that \mathbf=2^>. The
continuum 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, \mathbf=\aleph_1=\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 ...

Axiom of Choice
.
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 ...

real number line
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 ...

Giuseppe Peano
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 ...

cube
, 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 ...

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.


Geometry

Until the end of the 19th century, infinity was rarely discussed 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 figures. A mat ...

geometry
, 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 ...

line
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 ...

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 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 ...

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 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 ...

lines
intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately in classical geometry, while they need not to be distinguished in projective geometry. Before the use of
set theory 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 classical
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 figures. A mat ...

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 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

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 ...

physics
, 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 number


References


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