Extensive Definition
The precise origin of the infinity symbol ∞ is
unclear. One possibility is suggested by the name it is sometimes
called—the lemniscate, from the Latin
lemniscus, meaning "ribbon."
A popular explanation is that the infinity symbol
is derived from the shape of a Möbius
strip. Again, one can imagine walking along its surface
forever. However, this explanation is not plausible, since the
symbol had been in use to represent infinity for over two hundred
years before
August Ferdinand Möbius and Johann
Benedict Listing discovered the Möbius strip in 1858.
It is also possible that it is inspired by older
religious/alchemical symbolism. For instance, it
has been found in Tibetan rock carvings, and the
ouroboros, or infinity
snake, is often depicted in this shape.
John Wallis
is usually credited with introducing ∞ as a symbol for infinity in
1655 in his De
sectionibus conicis. One conjecture about why he chose this symbol
is that he derived it from a Roman
numeral for 1000 that was in turn derived from the Etruscan
numeral for 1000, which looked somewhat like CIƆ and was
sometimes used to mean "many." Another conjecture is that he
derived it from the Greek letter ω (omega), the last letter in the
Greek
alphabet.
Another possibility is that the symbol was chosen
because it was easy to rotate an "8" character by 90° when typesetting was done by
hand. The symbol is sometimes called a "lazy eight", evoking the
image of an "8" lying on its side.
Another popular belief is that the infinity
symbol is a clear depiction of the hourglass turned 90°.
Obviously, this action would cause the hourglass to take infinite
time to empty thus presenting a tangible example of infinity. The
invention of the hourglass predates the existence of the infinity
symbol allowing this theory to be plausible.
The infinity symbol is represented in Unicode by the
character ∞ (U+221E).
Mathematical infinity
Infinity is used in various branches of
mathematics.
Calculus
In real analysis, the symbol \infty, called "infinity", denotes an unbounded limit. x \rightarrow \infty means that x grows without bound, and x \rightarrow \infty means the value of x is decreasing without bound. If f(t) ≥ 0 for every t, then \int_^ \, f(t)\ dt \ = \infty means that f(t) does not bound a finite area from a to b
 \int_^ \, f(t)\ dt \ = \infty means that the area under f(t) is infinite.
 \int_^ \, f(t)\ dt \ = 1 means that the area under f(t) equals 1
Infinity is also used to describe infinite
series:
Algebraic properties
further Extended real number lineInfinity is often used not only to define a limit
but as a value in the affinely extended real number system. Points
labeled \infty and \infty can be added to the topological
space of the real numbers, producing the twopoint
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 onepoint
compactification of the real numbers, which is the real
projective line. Projective
geometry also introduces a line at
infinity in plane
geometry, and so forth for higher dimensions.
The extended real number line adds two elements
called infinity (\infty), greater than all other extended real
numbers, and negative infinity (\infty), less than all other
extended real numbers, for which some arithmetic operations may be
performed.
Complex analysis
As in real analysis, in complex
analysis the symbol \infty, called "infinity", denotes an
unbounded 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 giving the onepoint
compactification of the complex plane. When this is done, the
resulting space is a onedimensional complex
manifold, or Riemann
surface, called the extended complex plane or the Riemann
sphere. Arithmetic operations similar to those given below 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 z/0 = \infty for any
complex number z. In this context is often useful to consider
meromorphic
functions as maps into the Riemann sphere taking the value of
\infty at the poles. The domain of a complexvalued function may be
extended to include the point at infinity as well. One important
example of such functions is the group of Möbius
transformations.
Nonstandard analysis
The original formulation of the calculus by
Newton and Leibniz used infinitesimal quantities. In the twentieth
century, it was shown that this treatment could be put on a
rigorous footing through various logical
systems, including
smooth infinitesimal analysis and nonstandard
analysis. In the latter, infinitesimals are invertible, and
their inverses are infinite numbers. The infinities in this sense
are part of a whole field;
there is no equivalence between them as with the Cantorian transfinites. For example,
if H is an infinite number, then H + H = 2H and H + 1 are different
infinite numbers.
Set theory
A different type of "infinity" are the ordinal and cardinal infinities of set theory. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is alephnull (\aleph_0), the cardinality of the set of natural numbers. This modern mathematical conception of the quantitative infinite developed in the late nineteenth century from work by Cantor, Gottlob Frege, Richard Dedekind and others, using the idea of collections, or sets.Dedekind's approach was essentially to adopt the
idea of onetoone
correspondence as a standard for comparing the size of sets,
and to reject the view of Galileo (which derived from Euclid) that the
whole cannot be the same size as the part. 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.
Cantor defined two kinds of infinite numbers, the
ordinal
numbers and the cardinal
numbers. Ordinal numbers may be identified with wellordered
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 sequences which are maps from
the positive integers
leads to mappings
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 countably
infinite. 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
hyperreal
numbers, incorporate the ordinary (finite) numbers and infinite
numbers of different sizes.
Our intuition gained from finite sets
breaks down when dealing with infinite
sets. One example of this is
Hilbert's paradox of the Grand Hotel.
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 R
than natural numbers N. Namely, Cantor showed that \mathbf = 2^
> (see Cantor's
diagonal argument).
The continuum
hypothesis states that there is no cardinal
number between the cardinality of the reals and the cardinality
of the natural numbers, that is, \mathbf = \aleph_1 = \beth_1 (see
Beth
one). However, this hypothesis can neither be proved nor
disproved within the widely accepted
ZermeloFraenkel set theory, even assuming the Axiom of
Choice.
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 that this is equal to the number of points on a
plane and, indeed, in any finitedimensional space. These results
are highly counterintuitive, because they imply that there exist
proper
subsets of an infinite set S that have the same size as
S.
The first of these results is apparent by
considering, for instance, the
tangent function, which provides a onetoone
correspondence between the interval [0.5π, 0.5π] and R
(see also
Hilbert's paradox of the Grand Hotel). The second result was
proved by Cantor in 1878, but only became intuitively apparent in
1890, when Giuseppe
Peano introduced the spacefilling
curves, curved lines that twist and turn enough to fill the
whole of any square, or cube, or hypercube, or
finitedimensional space. These curves can be used to define a
onetoone
correspondence between the points in the side of a square and
those in the square.
Cantor also showed that sets with cardinality
strictly greater than \mathbf c exist (see his
generalized diagonal argument and theorem).
They include, for instance:

 the set of all subsets of R, i.e., the power set of R, written P(R) or 2R
 the set RR of all functions from R to R
Both have cardinality 2^\mathbf = \beth_2 >
\mathbf c (see Beth
two).
The
cardinal equalities \mathbf^2 = \mathbf, \mathbf c^ = \mathbf
c, and \mathbf c ^ = 2^ can be demonstrated using cardinal
arithmetic:
 \mathbf^2 = (2^)^2 = 2^ = 2^ = \mathbf,
 \mathbf c^ = (2^)^ = 2^ = 2^ = \mathbf,
 \mathbf c ^ = (2^)^ = 2^ = 2^.
Mathematics without infinity
Leopold
Kronecker rejected the notion of infinity and began a school of
thought, in the philosophy
of mathematics called finitism which influenced the
philosophical and mathematical school of mathematical
constructivism.
Physical infinity
In physics, approximations of
real
numbers are used for continuous
measurements and natural
numbers are used for discrete measurements (i.e.
counting). It is therefore assumed by physicists that no measurable quantity could
have an infinite value , for instance by taking an infinite value
in an
extended real number system (see also: hyperreal
number), or by requiring the counting of an infinite number of
events. It is for example presumed impossible for any body to have
infinite mass or infinite energy. There exists the concept of
infinite entities (such as an infinite plane wave)
but there are no means to generate such things.
It should be pointed out that this practice of
refusing infinite values for measurable quantities does not come
from
a priori 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.
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 infinite
series, unbounded functions,
etc., and may involve infinite quantities. Physicists however
require that the end result be physically meaningful. In quantum
field theory infinities arise which need to be interpreted in
such a way as to lead to a physically meaningful result, a process
called renormalization. One
application where infinities arise is the quantification of
thermodynamic temperatures.
However, there are some currentlyaccepted
circumstances where the end result is infinity. One example is
black
holes. The
general theory of relativity predicts that, when a star
experiences gravitational
collapse, it will eventually shrink down to a point of zero
size, and thus have infinite density. This is an example of what is
called a mathematical
singularity, or a point where the laws of mathematics, and
therefore of physics, break down. Some physicists now believe the
singularity may be physically real, and have since turned their
attention to finding new mathematics where infinities are
possible.
Infinity in cosmology
An intriguing question is whether actual infinity
exists in our physical universe: Are there infinitely
many stars? Does the universe have infinite volume? Does
space "go on forever"? This is an important open question of
cosmology.
Note that the question of being infinite is logically separate from
the question of having boundaries. The twodimensional surface of
the Earth, for example, is finite, yet has no edge. By
walking/sailing/driving/flying straight long enough, you'll return
to the exact spot you started from. The universe, at least in
principle, might have a similar topology; if you fly your space
ship straight ahead long enough, perhaps you would eventually
revisit your starting point. If, however,
the universe is ever expanding and your ship could not travel
faster than this rate of expansion then conceivably you would never
return to your starting point even on an infinite time scale since
your starting point would be receding away from you even as you
travel toward it.
Computer representations of infinity
The
IEEE floatingpoint standard specifies positive and negative
infinity values; these can be the result of arithmetic
overflow, division
by zero, or other exceptional operations.
Some programming
languages (for example, J
and
UNITY) specify greatest and
least elements, i.e. values
that compare (respectively) greater than or less than all other
values. These may also be termed top and bottom, or plus infinity
and minus infinity; they are useful as sentinel
values in algorithms involving sorting, searching or windowing.
In languages that do not have greatest and least elements, but do
allow overloading
of relational
operators, it is possible to create greatest and least elements
(with some overhead,
and the risk of incompatibility between implementations).
Perspective and points at infinity in the arts
Perspective
artwork utilizes the concept of imaginary vanishing
points, or points at
infinity, located at an infinite distance from the observer.
This allows artists to create paintings that 'realistically' depict
distance and foreshortening of objects. Artist M. C.
Escher is specifically known for employing the concept of
infinity in his work in this and other ways.
Notes
References
 The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity
 D. P. Agrawal (2000). Ancient Jaina Mathematics: an Introduction, Infinity Foundation.
 Exact Sciences from Jaina Sources
 L. C. Jain (1973). "Set theory in the Jaina school of mathematics", Indian Journal of History of Science.
 The Crest of the Peacock: NonEuropean Roots of Mathematics
 To Infinity and Beyond
 John J. O'Connor and Edmund F. Robertson (1998). 'Georg Ferdinand Ludwig Philipp Cantor', MacTutor History of Mathematics archive.
 John J. O'Connor and Edmund F. Robertson (2000). 'Jaina mathematics', MacTutor History of Mathematics archive.
 Ian Pearce (2002). 'Jainism', MacTutor History of Mathematics archive.
 Infinity and the Mind: The Science and Philosophy of the Infinite
 N. Singh (1988). 'Jaina Theory of Actual Infinity and Transfinite Numbers', Journal of Asiatic Society, Vol. 30.
 Everything and More: A Compact History of Infinity
See also
External links
 A Crash Course in the Mathematics of Infinite Sets, by Peter Suber. From the St. John's Review, XLIV, 2 (1998) 159. The standalone appendix to Infinite Reflections, below. A concise introduction to Cantor's mathematics of infinite sets.
 Infinite Reflections, 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) 159.
 Infinity, Principia Cybernetica
 Hotel Infinity
 The concepts of finiteness and infinity in philosophy
 Source page on medieval and modern writing on Infinity
 The Mystery Of The Aleph: Mathematics, the Kabbalah, and the Search for Infinity
illimitability in Arabic: لانهاية
illimitability in Aragonese: Infinito
illimitability in Bengali: অসীম
illimitability in Min Nan: Bûhān
illimitability in Bulgarian: Безкрайност
illimitability in Catalan: Infinit
illimitability in Czech: Nekonečno
illimitability in Welsh: Anfeidredd
illimitability in Danish: Uendelig
illimitability in German: Unendlichkeit
illimitability in Estonian: Lõpmatus
illimitability in Modern Greek (1453):
Άπειρο
illimitability in Spanish: Infinito
illimitability in Esperanto: Senfineco
illimitability in Basque: Infinitu
illimitability in French: Infini
illimitability in Galician: Infinito
illimitability in Korean: 무한
illimitability in Croatian: Beskonačnost
illimitability in Indonesian: Tak hingga
illimitability in Icelandic: Óendanleiki
illimitability in Italian: Infinito
(matematica)
illimitability in Hebrew: אינסוף
illimitability in Latin: Infinitas
illimitability in Lithuanian: Begalybė
illimitability in Lojban: ci'i
illimitability in Hungarian: Végtelen
illimitability in Marathi: अनंत
illimitability in Dutch: Oneindigheid
illimitability in Japanese: 無限
illimitability in Norwegian: Uendelig
illimitability in Norwegian Nynorsk:
Uendeleg
illimitability in Polish: Nieskończoność
illimitability in Portuguese: Infinito
illimitability in Russian: Бесконечность
illimitability in Albanian: Pafundësia
illimitability in Simple English: Infinity
illimitability in Slovak: Nekonečno
illimitability in Slovenian: Neskončnost
illimitability in Serbian: Бесконачност
illimitability in Finnish: Äärettömyys
illimitability in Swedish: Oändlighet
illimitability in Tamil: முடிவிலி
illimitability in Thai: อนันต์
illimitability in Vietnamese: Vô tận
illimitability in Ukrainian:
Нескінченність
illimitability in Chinese: 无穷