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For other uses of "Infinity" and "Infinite", see .
Infinity (symbol: ∞) is an abstract concept describing something without any bound or larger than any number. Philosophers have speculated about the nature of the infinite, such as , who proposed many paradoxes involving infinity, and , who used the idea of infinitely small quantities in his . Modern mathematics uses the concept of infinity in the solution of many practical and theoretical problems, such as in
and , and the idea also is used in
and the other sciences.
In mathematics, "infinity" is often treated as a
(i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as
formalized many ideas related to infinity and
during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called ). For example, the set of
is , while the infinite set of real numbers is .
Main article:
Ancient cultures had various ideas about the nature of infinity. The
did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.
The earliest recorded idea of infinity comes from , a
Greek philosopher who lived in . He used the word
which means infinite or limitless. However, the earliest attestable accounts of mathematical infinity come from
(c. 490 BCE? – c. 430 BCE?), a
Greek philosopher of southern Italy and member of the
School founded by .
called him the inventor of the . He is best known for his , described by
as "immeasurably subtle and profound".
In accordance with the traditional view of Aristotle, the
Greeks generally preferred to distin for example, instead of saying that there are an infinity of primes,
prefers instead to say that there are more prime numbers than contained in any given collection of
(, Book IX, Proposition 20).
However, recent readings of the
have hinted that Archimedes at least had an intuition about actual infinite quantities.
(c. 4th–3rd century BCE) classifies all numbers into three sets: , 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
In this work, two basic types of infinite numbers are distinguished. On both physical and
grounds, a distinction was made between
("countless, innumerable") and
("endless, unlimited"), between rigidly bounded and loosely bounded infinities.
European mathematicians started using infinite numbers in a systematic fashion in the 17th century.
first used the notation
{\displaystyle \infty }
for such a number, and exploited it in area calculations by dividing the region into
strips of width on the order of
{\displaystyle {\frac {1}{\infty }}}
used the notation
{\displaystyle i}
for an infinite number, and exploited it by applying the binomial formula to the
{\displaystyle i}
'th power, and infinite products of
{\displaystyle i}
wrote about equations with an infinite number of terms in his work .
Main article:
The infinity symbol
{\displaystyle \infty }
(sometimes called the ) is a mathematical symbol representing the concept of infinity. The symbol is encoded in
at U+221E ∞ INFINITY (HTML ∞ ·
as \infty.
It was introduced in 1655 by , and, since its introduction, has also been used outside mathematics in modern mysticism and literary .
, one of the co-inventors of , 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 .
In , the symbol
{\displaystyle \infty }
, called "infinity", is used to denote an unbounded .
{\displaystyle x\rightarrow \infty }
means that x grows without bound, and
{\displaystyle x\to -\infty }
means the value of x is decreasing without bound. If f(t) ≥ 0 for every t, then
{\displaystyle \int _{a}^{b}\,f(t)\ dt\ =\infty }
means that f(t) does not bound a finite area from
{\displaystyle a}
{\displaystyle b}
{\displaystyle \int _{-\infty }^{\infty }\,f(t)\ dt\ =\infty }
means that the area under f(t) is infinite.
{\displaystyle \int _{-\infty }^{\infty }\,f(t)\ dt\ =a}
means that the total area under f(t) is finite, and equals
{\displaystyle a}
Infinity is also used to describe :
{\displaystyle \sum _{i=0}^{\infty }\,f(i)=a}
means that the sum of the infinite series
to some real value
{\displaystyle a}
{\displaystyle \sum _{i=0}^{\infty }\,f(i)=\infty }
means that the sum of the infinite series
in the specific sense that the partial sums grow without bound.
Infinity can be used not only to define a limit but as a value in the extended real number system. Points labeled
{\displaystyle +\infty }
{\displaystyle -\infty }
can be added to the
of the real numbers, producing the two-point
of the real numbers. Adding algebraic properties to this gives us the . We can also treat
{\displaystyle +\infty }
{\displaystyle -\infty }
as the same, leading to the one-point compactification of the real numbers, which is the .
also refers to a
in plane geometry, a
in three-dimensional space, and so forth for higher .
By , the complex plane can be "wrapped" onto a sphere, with the top point of the sphere corresponding to infinity. This is called the .
the symbol
{\displaystyle \infty }
, called "infinity", denotes an unsigned infinite .
{\displaystyle x\rightarrow \infty }
means that the magnitude 
{\displaystyle |x|}
of x grows beyond any assigned value. A
can be added to the complex plane as a
giving the one-point
of the complex plane. When this is done, the resulting space is a one-dimensional , or , called the extended complex plane or the . Arithmetic operations similar to those given above for the extended real numbers can also be defined, though there is no distinction in the signs (therefore one exception is that infinity cannot be added to itself). On the other hand, this kind of infinity enables division by zero, namely
{\displaystyle z/0=\infty }
for any nonzero complex number z. In this context it is often useful to consider
as maps into the Riemann sphere taking the value of
{\displaystyle \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 .
Infinitesimals (ε) and infinites (ω) on the hyperreal number line (1/ε = ω/1)
The original formulation of
quantities. In the twentieth century, it was shown that this treatment could be put on a rigorous footing through various , including
and . In the latter, infinitesimals are invertible, and their inverses are infinite numbers. The infinities in this
there is no equivalence between them as with the Cantorian . For example, if H is an infinite number, then H + H = 2H and H + 1 are distinct infinite numbers. This approach to
is fully developed in .
Main articles:
One-to-one correspondence between infinite set and proper subset
A different form of "infinity" are the
infinities of set theory.
developed a system of , in which the first transfinite cardinal is
(?0), the cardinality of the set of . This modern mathematical conception of the quantitative infinite developed in the late nineteenth century from work by Cantor, ,
and others, using the idea of collections, or sets.
Dedekind's approach was essentially to adopt the idea of
as a standard for comparing the size of sets, and to reject the view of Galileo (which derived from ) that the whole cannot be the same size as the part (however, see
where he concludes that positive integers which are squares and all positive integers are the same size). An infinite set can simply be defined as one having the same size as at lea this notion of infinity is called . The diagram 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 numbers and cardinal numbers. Ordinal numbers may be identified with
sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. Generalizing finite and the ordinary infinite
which are maps from the positive
from ordinal numbers, and 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 . 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.[] Certain extended number systems, such as the , incorporate the ordinary (finite) numbers and infinite numbers of different sizes.
Main article:
One of Cantor's most important results was that the cardinality of the continuum
{\displaystyle \mathbf {c} }
is greater than that of the natural numbers
{\displaystyle {\aleph _{0}}}
; that is, there are more real numbers R than natural numbers N. Namely, Cantor showed that
{\displaystyle \mathbf {c} =2^{\aleph _{0}}&{\aleph _{0}}}
states that there is no
between the cardinality of the reals and the cardinality of the natural numbers, that is,
{\displaystyle \mathbf {c} =\aleph _{1}=\beth _{1}}
(see ). However, this hypothesis can neither be proved nor disproved within the widely accepted , even assuming the .
can be used to show not only that the number of points in a
is equal to the number of points in any
of that line, but that this is equal to the number of points on a plane and, indeed, in any
The first three steps of a fractal construction whose limit is a , showing that there are as many points in a one-dimensional line as in a two-dimensional square.
The first of these results is apparent by considering, for instance, the
function, which provides a
between the
(-π/2, π/2) and R (see also ). The second result was proved by Cantor in 1878, but only became intuitively apparent in 1890, when
introduced the space-filling curves, curved lines that twist and turn enough to fill the whole of any square, or , or , or finite-dimensional space. These curves can be used to define a one-to-one correspondence between the points in the side of a square and those in the square.
Main article:
Infinite- spaces are widely used in
and , particularly as , such as . Common examples are the infinite-dimensional
and the infinite-dimensional
K(Z/2Z,1).
The structure of a
object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters—some with infinite, and others with finite surface areas. One such fractal curve with an infinite perimeter and finite surface area is the .
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
called , an extreme form of the philosophical and mathematical schools of
This section does not
any . Please help improve this section by . Unsourced material may be challenged and . (December 2009) ()
In , approximations of
are used for
measurements and
are used for
measurements (i.e. counting). It is therefore assumed by physicists that no
could have an infinite value[]. For instance, by taking an infinite value in an
system, or by requiring the counting of an infinite number of events. It is, for example, presumed impossible for any type of body to have infinite mass or infinite energy. Concepts of infinite things such as an infinite
exist, but there are no experimental means to generate them.
The practice of refusing infinite values for measurable quantities does not come from
or ideological motivations, but rather from more methodological and pragmatic motivations[ – ][]. One of the needs of any physical and scientific theory is to give usable formulas that correspond to or at least approximate reality. As an example, if any object of infinite gravitational mass were to exist, any usage of the formula to calculate the gravitational force would lead to an infinite result, which would be of no benefit since the result would be always the same regardless of the position and the mass of the other object. The formula would be useful neither to compute the force between two objects of finite mass nor to compute their motions. If an infinite mass object were to exist, any object of finite mass would be attracted with infinite force (and hence acceleration) by the infinite mass object, which is not what we can observe in reality. Sometimes infinite result of a physical quantity may mean that the theory being used to compute the result may be approaching the point where it fails. This may help to indicate the limitations of a theory.
This point of view does not mean that infinity cannot be used in physics. For convenience's sake, calculations, equations, theories and approximations often use , unbounded , etc., and may involve infinite quantities. Physicists however require that the end result be physically meaningful. In
infinities arise which need to be interpreted in such a way as to lead to a physically meaningful result, a process called .
However, there are some theoretical circumstances where the end result is infinity. One example is the
in the description of . Some solutions of the equations of the
allow for finite mass distributions of zero size, and thus infinite density. This is an example of what is called a , or a point where a physical theory breaks down. This does not necessarily mean that physi it may mean simply that the theory is incapable of describing the situation properly. Two other examples occur in inverse-square force laws of the gravitational force equation of
of electrostatics. At r=0 these equations evaluate to infinities.
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
proposed an unbounded universe in On the Infinite Universe and Worlds: "In 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."
have long sought to discover whether infinity exists in our physical : Are there an infinite number of stars? Does the universe have infinite volume? Does space ? This is an open question of . 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 . 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
in the spectrum of the . As to date, analysis of the radiation patterns recorded by the
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
hypothesis, which, when explained by astrophysicists such as , posits that there are an infinite number and variety of universes.
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."
standard (IEEE 754) specifies the positive and negative infinity values. These are defined as the result of , , and other exceptional operations.
Some , such as
and , allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as , as they compare (respectively) greater than or less than all other values. They have uses as
involving , , or .
In languages that do not have greatest and least elements, but do allow
of , 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 , the infinity values may still be accessible and usable as the result of certain operations.
artwork utilizes the concept of , roughly corresponding to mathematical , located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms. Artist
is specifically known for employing the concept of infinity in his work in this and other ways.
considers the concept of infinity in mathematics and the sciences as a metaphor. This perspective is based on the basic metaphor of infinity (BMI), defined as the ever-increasing sequence &1,2,3,...&.
The symbol is often used romantically to represent eternal love. Several types of jewelry are fashioned into the infinity shape for this purpose.[]
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in Wiktionary, the free dictionary.
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at the . ()
, by Peter Suber. From the St. John's Review, XLIV, 2 (. 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 (.
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Ian Pearce (2002). , MacTutor History of Mathematics archive.
(compilation of articles about infinity in physics, mathematics, and philosophy)
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