infinity

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Infinity is that which is boundless, endless, or larger than any
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
. It is often denoted by the
infinity symbol The infinity symbol (\infty) is a List of mathematical symbols, mathematical symbol representing the concept of infinity. This symbol is also called a lemniscate, after the lemniscate curves of a similar shape studied in algebraic geometry, or ...
. Since the time of the
ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean Sea, Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, classical antiquity ( AD 600), th ...
, 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 infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, mathematicians began to work with
infinite series In mathematics, a series is, roughly speaking, a description of the operation of addition, adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalizat ...
and what some mathematicians (including l'Hôpital and 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 played a pivotal role in the creation of set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, ...
enlarged the mathematical study of infinity by studying
infinite set In 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 bran ...
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, the cardinality of a set (mathematics), set is a measure of the number of Element (mathematics), elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19 ...
of the line) is larger than the number of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
s. In this usage, infinity is a mathematical concept, and infinite
mathematical object A mathematical object is an abstract concept arising in mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantit ...
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, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
, on which most of modern mathematics can be developed, is the
axiom of infinity In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the ...
, 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 primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely related to many other ar ...
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 The operators in elementary arithmetic Arithmetic () is an elementary part of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are ...
. In
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
and
cosmology Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount (lexicographer), Thomas Blount's ''Glossographia'', and in 1731 taken up in ...
, whether the Universe is spatially infinite is an open question.

History

Ancient cultures had various ideas about the nature of infinity. The ancient Indians and the
Greeks The Greeks or Hellenes (; el, Έλληνες, ''Éllines'' ) are an ethnic group and nation indigenous to the Eastern Mediterranean and the Black Sea regions, namely Greece, Greek Cypriots, Cyprus, Greeks in Albania, Albania, Greeks in Italy, ...
did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.

Early Greek

The earliest recorded idea of infinity in Greece may be that of
Anaximander Anaximander (; grc-gre, Ἀναξίμανδρος ''Anaximandros''; ) was a Pre-Socratic philosophy, pre-Socratic Ancient Greek philosophy, Greek philosopher who lived in Miletus,"Anaximander" in ''Chambers's Encyclopædia''. London: George Newn ...
(c. 610 – c. 546 BC) a
pre-Socratic Pre-Socratic philosophy, also known as early Greek philosophy, is ancient Greek philosophy before Socrates. Pre-Socratic philosophers were mostly interested in cosmology, the beginning and the substance of the universe, but the inquiries of thes ...
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, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These might include the set of natural numbers, extend ...
'', 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 In Classical antiquity, the Hellenistic period covers the time in History of the Mediterranean region, Mediterranean history after Classical Greece, between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as sig ...
Greeks had a "horror of the infinite" which would, for example, explain why
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
(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 "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: 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

Zeno of Elea Zeno of Elea (; grc, wikt:Ζήνων, Ζήνων ὁ Ἐλεᾱ́της; ) was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He i ...
( 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 mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...
as "immeasurably subtle and profound".
Achilles In Greek mythology, Achilles ( ) or Achilleus ( grc-gre, wikt:Ἀχιλλεύς, Ἀχιλλεύς) was a hero of the Trojan War, the greatest of all the Greek warriors, and the central character of Homer's ''Iliad''. He was the son of the Ner ...
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 Eleatics 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 Nobility is a social class found in many societies that have an aristocracy (class), aristocracy. It is normally ranked immediately below Royal family, royalty. Nobility has often been an Estates o ...
provided both a satisfactory definition of a limit and a proof that, for , $a+ax+ax^2+ax^3+ax^4+ax^5+\cdots=\frac.$ Suppose that Achilles is running at 10 meters per second, the tortoise is walking at 0.1 meters per second, and the latter has a 100-meter head start. The duration of the chase fits Cauchy's pattern with 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 who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament of the United Kingdom, ...
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, an infinitesimal number is a quantity that is closer to 0, zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century New Latin, Modern Latin coinage ''infinitesimus'', which ori ...
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, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
wrote about equations with an infinite number of terms in his work '' De analysi per aequationes numero terminorum infinitas''.

Mathematics

Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
opened a mathematico-philosophic address given in 1930 with:

Symbol

The infinity symbol $\infty$ (sometimes called the lemniscate) is a mathematical symbol representing the concept of infinity. The symbol is encoded in
Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, ...
at and in
LaTeX Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well. In nature, latex is found as a wikt:milky, milky fluid found in 10% of all flowering plants ( ...
as \infty. It was introduced in 1655 by
John Wallis John Wallis (; la, Wallisius; ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament of the United Kingdom, ...
, 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 A word is a basic element of language Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the pri ...
.

Calculus

Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathema ...
, one of the co-inventors of
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, 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.

Real analysis

In
real analysis In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
, the symbol $\infty$, called "infinity", is used to denote an unbounded 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 $f\left(t\right)\ge 0$ for every ''$t$'', then * $\int_^ f\left(t\right)\, dt = \infty$ means that $f\left(t\right)$ does not bound a finite area from $a$ to $b.$ * $\int_^ f\left(t\right)\, dt = \infty$ means that the area under $f\left(t\right)$ is infinite. * $\int_^ f\left(t\right)\, dt = a$ means that the total area under $f\left(t\right)$ is finite, and is equal to $a.$ Infinity can also be used to describe
infinite series In mathematics, a series is, roughly speaking, a description of the operation of addition, adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalizat ...
, as follows: * $\sum_^ f\left(i\right) = a$ means that the sum of the infinite series converges to some real value $a.$ * $\sum_^ f\left(i\right) = \infty$ 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, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
of the real numbers, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat $+\infty$ and $-\infty$ as the same, leading to the one-point compactification of the real numbers, which is the 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, pro ...
also refers to a
line at infinity In geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of ...
in plane geometry, a plane at infinity in three-dimensional space, and a
hyperplane at infinity In geometry, any hyperplane ''H'' of a projective space ''P'' may be taken as a hyperplane at infinity. Then the set complement is called an affine space. For instance, if are homogeneous coordinates for ''n''-dimensional projective space, then t ...
for general
dimensions In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
, each consisting of points at infinity.

Complex analysis

In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
the symbol $\infty$, called "infinity", denotes an unsigned infinite limit. $x \rightarrow \infty$ 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, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
giving the one-point compactification of the complex plane. When this is done, the resulting space is a one-dimensional
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas (topology), atlas of chart (topology), charts to the open unit disc in \mathbb^n, such that the transition maps are Holomorphic function, holomorphic. ...
, or
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versio ...
, called the extended complex plane or the
Riemann sphere In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
. 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, division by zero is division (mathematics), division where the divisor (denominator) is 0, zero. Such a division can be formally expression (mathematics), expressed as \tfrac, where is the dividend (numerator). In ordinary ari ...
, 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 set, 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 is ...
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 transformations (see Möbius transformation § Overview).

Nonstandard analysis

The original formulation of
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
and Gottfried Leibniz used
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to 0, zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century New Latin, Modern Latin coinage ''infinitesimus'', which ori ...
quantities. In the second half of the 20th century, it was shown that this treatment could be put on a rigorous footing through various
logical system A formal system is an abstract structure 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 form ...
s, including smooth infinitesimal analysis 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 and
cardinal Cardinal or The Cardinal may refer to: Animals * Cardinal (bird) or Cardinalidae, a family of North and South American birds **''Cardinalis'', genus of cardinal in the family Cardinalidae **''Cardinalis cardinalis'', or northern cardinal, the ...
infinities of set theory—a system of
transfinite number In mathematics, transfinite numbers are numbers that are "Infinity, infinite" in the sense that they are larger than all finite set, finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardina ...
s first developed by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, ...
. In this system, the first transfinite cardinal is
aleph-null In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named af ...
(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 ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
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, Mathematical logic, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the fath ...
,
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathemati ...
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 is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
as a standard for comparing the size of sets, and to reject the view of Galileo (derived from
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
) that the whole cannot be the same size as the part. (However, see Galileo's paradox where Galileo concludes that positive integers cannot be compared to the subset of positive square integers since both are infinite sets.) An infinite set can simply be defined as one having the same size as at least one of its proper 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, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least n ...
s and
cardinal number In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
s. Ordinal numbers characterize well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and (ordinary) infinite
sequence In mathematics, a sequence is an enumerated collection of mathematical object, objects in which repetitions are allowed and order theory, order matters. Like a Set (mathematics), set, it contains Element (mathematics), members (also called ''eleme ...
s which are maps from the positive
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
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, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
. 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 is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
''. 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, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
states that there is no
cardinal number In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
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, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
, 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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
can be used to show not only that the number of points in a
real number line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
is equal to the number of points in any segment of that line, but also that this is equal to the number of points on a plane and, indeed, in any
finite-dimensional In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
space. The first of these results is apparent by considering, for instance, the
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given Point (geometry), point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitesimal, infinitely ...
function, which provides a
one-to-one correspondence In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
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 in their work, typically to solve mathematical problems. Mathematicians are concer ...
introduced the
space-filling curve In mathematical analysis, a space-filling curve is a curve whose Range of a function, range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first t ...
s, curved lines that twist and turn enough to fill the whole of any square, or
cube In geometry, a cube is a three-dimensional space, three-dimensional solid object bounded by six square (geometry), square faces, Facet (geometry), facets or sides, with three meeting at each vertex (geometry), vertex. Viewed from a corner it i ...
, or
hypercube In geometry, a hypercube is an N-dimensional space, ''n''-dimensional analogue of a Square (geometry), square () and a cube (). It is a Closed set, closed, Compact space, compact, Convex polytope, convex figure whose 1-N-skeleton, skeleton consis ...
, 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 (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
, except in the context of processes that could be continued without any limit. For example, a
line Line most often refers to: * Line (geometry) In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclide ...
was what is now called a
line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct end Point (geometry), points, and contains every point on the line that is between its endpoints. The length of a line segment is give ...
, 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 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, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These might include the set of natural numbers, extend ...
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, pro ...
, where points at infinity are added to the
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
for modeling the perspective effect that shows
parallel lines In geometry, parallel lines are coplanar straight line (geometry), lines that do not intersecting lines, intersect at any point. Parallel planes are plane (geometry), planes in the same three-dimensional space that never meet. ''Parallel curve ...
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, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, paral ...
, two distinct
lines Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
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, ...
for the foundation of mathematics, points and lines were viewed as distinct entities, and a point could be ''located on a line''. With the universal use of set theory in mathematics, the point of view has dramatically changed: a line is now considered as ''the set of its points'', and one says that a point ''belongs to a line'' instead of ''is located on a line'' (however, the latter phrase is still used). In particular, in modern mathematics, lines are ''infinite sets''.

Infinite dimension

The
vector space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
s that occur in classical
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
have always a finite
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
, 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 Functional analysis is a branch of mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, ...
where
function space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mo ...
s are generally vector spaces of infinite dimension. In topology, some constructions can generate
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s of infinite dimension. In particular, this is the case of iterated loop spaces.

Fractals

The structure of a
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...
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 A fractal curve is, loosely, a mathematical curve (mathematics), curve whose shape retains the same general pattern of Pathological (mathematics), irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fract ...
with an infinite perimeter and finite area is the
Koch snowflake The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curv ...
.

Mathematics without infinity

Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a Germans, German mathematician who worked on number theory, abstract algebra, algebra and mathematical logic, logic. He criticized Georg Cantor's work on set theory, and was quoted b ...
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 The philosophy of mathematics is the branch of philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such quest ...
called finitism, an extreme form of mathematical philosophy in the general philosophical and mathematical schools of
constructivism Constructivism may refer to: Art and architecture * Constructivism (art), an early 20th-century artistic movement that extols art as a practice for social purposes * Constructivist architecture, an architectural movement in Russia in the 1920s a ...
and
intuitionism In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fu ...
.

Physics

In
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
, approximations of
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s are used for 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 ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s are used for discrete measurements (i.e., counting). Concepts of infinite things such as an infinite
plane wave In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science ...
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 Giordano Bruno (; ; la, Iordanus Brunus Nolanus; born Filippo Bruno, January or February 1548 – 17 February 1600) was an Italian people, Italian philosopher, mathematician, poet, cosmological theorist, and Hermetica, Hermetic occultist. He i ...
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."
Cosmologists Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...
have long sought to discover whether infinity exists in our physical
universe The universe is all of space and time and their contents, including planets, stars, galaxy, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmology, cosmological description of the development of ...
: Are there an infinite number of stars? Does the universe have infinite volume? Does space " go on forever"? This is still an open question of
cosmology Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount (lexicographer), Thomas Blount's ''Glossographia'', and in 1731 taken up in ...
. 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 In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
. 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 background radiation Cosmic background radiation is electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromag ...
. To date, analysis of the radiation patterns recorded by the
WMAP The Wilkinson Microwave Anisotropy Probe (WMAP), originally known as the Microwave Anisotropy Probe (MAP and Explorer 80), was a NASA spacecraft operating from 2001 to 2010 which measured temperature differences across the sky in the cosmic mic ...
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 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 The multiverse is a Hypothesis, hypothetical group of multiple universes. Together, these universes comprise everything that exists: the entirety of space, time, matter, energy, information, and the physical laws and Physical constant, consta ...
hypothesis, which, when explained by astrophysicists such as Michio Kaku, posits that there are an infinite number and variety of universes. Also,
cyclic model A cyclic model (or oscillating model) is any of several cosmological models in which the universe follows infinite, or indefinite, self-sustaining cycles. For example, the oscillating universe theory briefly considered by Albert Einstein in 1930 t ...
s posit an infinite amount of
Big Bang The Big Bang event is a physical theory that describes how the Expansion of the universe, universe expanded from an initial state of high Energy density, density and temperature. Various Physical cosmology, cosmological models of the Big Ba ...
s, resulting in an infinite variety of universes after each Big Bang event in an infinite cycle.

Logic

In
logic Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science investigating h ...
, an
infinite regress An infinite regress is an infinite series of entities governed by a recursive principle that determines how each entity in the series depends on or is produced by its predecessor. In the epistemic regress, for example, a belief is justified beca ...
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 indefinite values). These are defined as the result of
arithmetic overflow Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operation (mathematics), operations on numbers—addition, subtraction, multiplication, division (mathematics), division, exponent ...
,
division by zero In mathematics, division by zero is division (mathematics), division where the divisor (denominator) is 0, zero. Such a division can be formally expression (mathematics), expressed as \tfrac, where is the dividend (numerator). In ordinary ari ...
, and other exceptional operations. Some
programming language A programming language is a system of notation for writing computer program, computer programs. Most programming languages are text-based formal languages, but they may also be visual programming language, graphical. They are a kind of computer ...
s, such as
Java Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's List ...
and J, allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least elements, as they compare (respectively) greater than or less than all other values. They have uses as
sentinel value In computer programming, a sentinel value (also referred to as a flag value, trip value, rogue value, signal value, or dummy data) is a special value (computer science), value in the context of an algorithm which uses its presence as a condition of ...
s in
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
s involving
sorting Sorting refers to ordering data in an increasing or decreasing manner according to some linear relationship among the data items. # Collating order, ordering: arranging items in a sequence ordered by some criterion; # categorization, categorizing ...
,
searching Searching or search may refer to: Computing technology * Search algorithm, including keyword search ** :Search algorithms * Search and optimization for problem solving in artificial intelligence * Search engine technology, software for finding ...
, or windowing. In languages that do not have greatest and least elements, but do allow overloading of
relational operator In computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied sc ...
s, it is possible for a programmer to ''create'' the greatest and least elements. In languages that do not provide explicit access to such values from the initial state of the program, but do implement the floating-point
data type In computer science and computer programming, a data type (or simply type) is a set of possible values and a set of allowed operations on it. A data type tells the compiler or Interpreter (computing), interpreter how the programmer intends to u ...
, the infinity values may still be accessible and usable as the result of certain operations. In programming, an
infinite loop In computer programming, an infinite loop (or endless loop) is a sequence of instructions that, as written, will continue endlessly, unless an external intervention occurs ("pull the plug"). It may be intentional. Overview This differs from: * ...
is a loop whose exit condition is never satisfied, thus executing indefinitely.

Arts, games, and cognitive sciences

Perspective artwork uses the concept of
vanishing point A vanishing point is a point (geometry), point on the projection plane, image plane of a graphical perspective, perspective drawing where the two-dimensional perspective projections of mutually parallel (geometry), parallel lines in three-dimen ...
s, roughly corresponding to mathematical 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 Chess is a board game between two Player (game), players. It is sometimes called international chess or Western chess to distinguish it from chess variant, related games, such as xiangqi (Chinese chess) and shogi (Japanese chess). The current ...
played on an unbounded board are called
infinite chess Infinite chess is any Chess variant, variation of the game of chess played on an Bounded set, unbounded chessboard. Versions of infinite chess have been introduced independently by multiple players, chess theorists, and mathematicians, both as a ...
. 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,...>.

* 0.999... *
Aleph number In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named af ...
* Ananta *
Exponentiation Exponentiation is a mathematics, mathematical operation (mathematics), operation, written as , involving two numbers, the ''Base (exponentiation), base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". W ...
*
Indeterminate form In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limit (mathematics), limits; if the expression o ...
*
Infinite monkey theorem The infinite monkey theorem states that a monkey hitting keys at randomness, random on a typewriter keyboard for an infinity, infinite amount of time will almost surely type any given text, such as the complete works of William Shakespeare. In f ...
*
Infinite set In 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 bran ...
*
Infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to 0, zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century New Latin, Modern Latin coinage ''infinitesimus'', which ori ...
Surreal number In mathematics, the surreal number system is a total order, totally ordered proper class containing the real numbers as well as Infinity, infinite and infinitesimal, infinitesimal numbers, respectively larger or smaller in absolute value than any ...

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 The MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland. It contains detailed biography, biographies on many historical and contemporar ...
''. * 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 The MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland. It contains detailed biography, biographies on many historical and contemporar ...
''. * John J. O'Connor and Edmund F. Robertson (2000)
'Jaina mathematics'
, ''MacTutor History of Mathematics archive''. * Ian Pearce (2002)

''MacTutor History of Mathematics archive''.

* ttp://dictionary.of-the-infinite.com Dictionary of the Infinite(compilation of articles about infinity in physics, mathematics, and philosophy) {{Authority control Concepts in logic Philosophy of mathematics Mathematical objects