![Reflections 1090029](https://upload.wikimedia.org/wikipedia/commons/9/9e/Reflections_1090029.jpg)
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 .
Since the time of the
ancient Greeks
Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cultu ...
, 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 mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, mathematicians began to work with
infinite series 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, which has become a fundamental theory in mathematics. Cantor established the importance of ...
enlarged the mathematical study of infinity by studying
infinite sets 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 is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
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 ...
s. In this usage, infinity is a mathematical concept, and infinite
mathematical objects 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 such as ...
, on which most of modern mathematics can be developed, is the
axiom of infinity, 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 structures. It is closely related to many other areas of mathematics and has many appl ...
that may seem to have nothing to do with them. For example,
Wiles's proof of
Fermat's Last Theorem implicitly relies on the existence of
very large infinite sets for solving a long-standing problem that is stated in terms of
elementary arithmetic.
In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
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, Cyprus, Albania, Italy, Turkey, Egypt, and, to a lesser extent, oth ...
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 Greek philosopher who lived in Miletus,"Anaximander" in ''Chambers's Encyclopædia''. London: George Newnes, 1961, Vol. 1, p. 403. a city of Ionia (in moder ...
(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'', 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 Mediterranean history after Classical Greece, between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as signified by the Battle of Actium in ...
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's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proved by Euclid in his work '' Elements''. There are several proofs of the theorem.
Euclid's proof
Euclid offere ...
, Euclid "was the first to overcome the horror of the infinite". There is a similar controversy concerning Euclid's
parallel postulate, 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, 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 is best known fo ...
( 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, ...
as "immeasurably subtle and profound".
Achilles
In Greek mythology, Achilles ( ) or Achilleus ( grc-gre, 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 Nereid Thetis and Peleus, k ...
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 group of pre-Socratic philosophers in the 5th century BC centered around the ancient Italian Greek colony of Elea ( grc, Ἐλέα), located in present-day Campania in southern Italy.
The primary philosophers who are associat ...
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 Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
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 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
Early Indian
The
Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets:
enumerable
An enumeration is a complete, ordered List (information), listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the element (mathematics), elements of a Set (mathem ...
, 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 and, later, the royal ...
first used the notation
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 zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
strips of width on the order of
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, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
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 m ...
''.
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
(sometimes called the
lemniscate
In algebraic geometry, a lemniscate is any of several figure-eight or -shaped curves. The word comes from the Latin "''lēmniscātus''" meaning "decorated with ribbons", from the Greek λημνίσκος meaning "ribbons",. or which alternativel ...
) 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 Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expre ...
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 milky fluid found in 10% of all flowering plants (angiosperms ...
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 and, later, the royal ...
, 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 that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conce ...
.
Calculus
Gottfried Leibniz, one of the co-inventors of
infinitesimal calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, 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, the symbol
, called "infinity", is used to denote an unbounded
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
. The notation
means that ''
'' increases without bound, and
means that ''
'' decreases without bound. For example, if
for every ''
'', then
*
means that
does not bound a finite area from
to
*
means that the area under
is infinite.
*
means that the total area under
is finite, and is equal to
Infinity can also be used to describe
infinite series, as follows:
*
means that the sum of the infinite series
converges to some real value
*
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
and
can be added to the
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
of the real numbers, producing the two-point
compactification
Compactification may refer to:
* Compactification (mathematics), making a topological space compact
* Compactification (physics), the "curling up" of extra dimensions in string theory
See also
* Compaction (disambiguation)
Compaction may refer t ...
of the real numbers. Adding algebraic properties to this gives us the
extended real numbers. We can also treat
and
as the same, leading to the
one-point compactification of the real numbers, which is the
real projective line
In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line (geometry), line that has been historically introduced to solve a problem set by visual perspective (visual), perspect ...
.
Projective geometry also refers to a
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 contained in the hyperplane at infinity of any projective space of higher dimension. This article will be concerned so ...
in three-dimensional space, and a
hyperplane at infinity for general
dimensions, each consisting of
points at infinity.
Complex analysis
![Riemann sphere1](https://upload.wikimedia.org/wikipedia/commons/3/32/Riemann_sphere1.svg)
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
, called "infinity", denotes an unsigned infinite
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
.
means that the magnitude
of ''
'' grows beyond any assigned value. A
point labeled can be added to the complex plane as a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
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 of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a com ...
, 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 vers ...
, called the extended complex plane or the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...
. Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (which leads to the one exception that infinity cannot be added to itself). On the other hand, this kind of infinity enables
division by zero, namely
for any nonzero complex number ''
''. In this context, it is often useful to consider
meromorphic functions as maps into the Riemann sphere taking the value of
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 and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' ...
s (see
Möbius transformation § Overview).
Nonstandard analysis
![Números hiperreales](https://upload.wikimedia.org/wikipedia/commons/5/53/N%C3%BAmeros_hiperreales.png)
The original formulation of
infinitesimal calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
by
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
and Gottfried Leibniz used
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
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
Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of bein ...
and
nonstandard analysis. In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this sense are part of a
hyperreal field
In mathematics, the system of hyperreal numbers is a way of treating Infinity, infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an Field extension, extension of the real numbe ...
; 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
In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered m ...
is fully developed in .
Set theory
![Infinity paradoxon - one-to-one correspondence between infinite set and proper subset](https://upload.wikimedia.org/wikipedia/commons/a/ab/Infinity_paradoxon_-_one-to-one_correspondence_between_infinite_set_and_proper_subset.gif)
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 numbers 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, which has become a fundamental theory in mathematics. Cantor established the importance of ...
. In this system, the first transfinite cardinal is
aleph-null (
ℵ0), 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, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic phil ...
,
Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...
and others—using the idea of collections or sets.
Dedekind's approach was essentially to adopt the idea of
one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (derived from
Euclid
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
In mathematics, a set ''A'' is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset ''B'' of ''A'' is equinumerous to ''A''. Explicitly, this means that there exists a bijective function from ''A'' ont ...
. 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, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
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 objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s which are maps from the positive
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, a set is countable if either it is 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 from it into the natural numbers; ...
. If a set is too large to be put in one-to-one correspondence with the positive integers, it is called ''
uncountable''. 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
is greater than that of the natural numbers
; that is, there are more real numbers than natural numbers . Namely, Cantor showed that
.
The
continuum hypothesis states that there is no
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
between the cardinality of the reals and the cardinality of the natural numbers, that is,
.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 such as ...
, 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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
.
Cardinal arithmetic can be used to show not only that the number of points in a
real number line is equal to the number of points in any
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, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...
space.
![Peanocurve](https://upload.wikimedia.org/wikipedia/commons/6/64/Peanocurve.svg)
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 is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
function, which provides a
one-to-one correspondence between the
interval () and.The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when
Giuseppe Peano introduced the
space-filling curve
In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an ''n''-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space ...
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 solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the only r ...
, or
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...
, 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 c ...
, except in the context of processes that could be continued without any limit. For example, a
line
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 ...
was what is now called a
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
, 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 was
projective geometry, 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 intersecting "at infinity". Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a
projective plane, two distinct
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 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, is mostly conce ...
for the
foundation of mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...
, 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 and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
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 c ...
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, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
where
function space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
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 geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s of infinite dimension. In particular, this is the case of
iterated loop space
In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topolog ...
s.
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 illu ...
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
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's ...
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 fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, approximations of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s are used for
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** 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
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
measurements (i.e., counting). Concepts of infinite things such as an infinite
plane wave
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position \vec x in space and any time t, th ...
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 philosopher, mathematician, poet, cosmological theorist, and Hermetic occultist. He is known for his cosmologic ...
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 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, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
: 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 from the Big Bang. The origin of this radiation depends on the region of the spectrum that is observed. One component is the cosmic microwave background. This component is redshifted pho ...
. 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 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 constants that describe them. The di ...
hypothesis, which, when explained by astrophysicists such as
Michio Kaku, posits that there are an infinite number and variety of universes. Also,
cyclic models posit an infinite amount of
Big Bang
The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
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 formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
, 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
Indefinite may refer to:
* the opposite of definite in grammar
** indefinite article
** indefinite pronoun
* Indefinite integral, another name for the antiderivative
* Indefinite forms in algebra, see definite quadratic forms
* an indefinite matr ...
values). These are defined as the result of
arithmetic overflow,
division by zero, and other exceptional operations.
Some
programming language
A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language.
The description of a programming ...
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
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an ele ...
, as they compare (respectively) greater than or less than all other values. They have uses as
sentinel values 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.
# ordering: arranging items in a sequence ordered by some criterion;
# categorizing: grouping items with similar pro ...
,
searching, or
windowing.
In languages that do not have greatest and least elements, but do allow
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
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 how the programmer intends to use the data. Most progra ...
, the infinity values may still be accessible and usable as the result of certain operations.
In programming, an
infinite loop is a
loop
Loop or LOOP may refer to:
Brands and enterprises
* Loop (mobile), a Bulgarian virtual network operator and co-founder of Loop Live
* Loop, clothing, a company founded by Carlos Vasquez in the 1990s and worn by Digable Planets
* Loop Mobile, an ...
whose exit condition is never satisfied, thus executing indefinitely.
Arts, games, and cognitive sciences
Perspective artwork uses the concept of
vanishing points, 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 for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to disti ...
played on an unbounded board are called
infinite chess
Infinite chess is any variation of the game of chess played on an unbounded chessboard. Versions of infinite chess have been introduced independently by multiple players, chess theorists, and mathematicians, both as a playable game and as a mod ...
.
Cognitive scientist George Lakoff
George Philip Lakoff (; born May 24, 1941) is an American cognitive linguistics, cognitive linguist and philosopher, best known for his thesis that people's lives are significantly influenced by the conceptual metaphors they use to explain comple ...
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,...>.
See also
*
0.999...
*
Aleph number
*
Ananta
*
Exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
*
Indeterminate form
*
Infinite monkey theorem
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, such as the complete works of William Shakespeare. In fact, the monkey would ...
*
Infinite set
*
Infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
*
Paradoxes of infinity
This list includes well known paradoxes, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. This list collects only scenarios that have been called a paradox by at least one source and have their ...
*
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
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 biographies on many historical and contemporary mathemati ...
''.
* 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
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 biographies on many historical and contemporary mathemati ...
''.
* 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)
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Concepts in logic
Philosophy of mathematics
Mathematical objects