Infinity (symbol:
∞
{\displaystyle \infty }
) is a concept describing something without
any bound, or something larger than any natural
number. Philosophers have speculated about
the nature of the infinite, for example Zeno
of Elea, who proposed many paradoxes involving
infinity, and Eudoxus of Cnidus, who used
the idea of infinitely small quantities in
his method of exhaustion. This idea is also
at the basis of infinitesimal calculus.
At the end of 19th century, Georg Cantor introduced
and studied infinite sets and infinite numbers,
which are now an essential part of the foundation
of mathematics. For example, in modern mathematics,
a line is commonly viewed as the set of all
its points, and their infinite number (the
cardinality of the line) is larger than the
number of integers. Thus the mathematical
concept of infinity refines and extends the
old philosophical concept. It is used everywhere
in mathematics, even in areas such as combinatorics
and number theory that may seem to have nothing
to do with it. For example, Wiles's proof
of Fermat's Last Theorem uses the existence
of very large infinite sets.
The concept of infinity is also used in physics
and the other sciences.
== History ==
Ancient cultures had various ideas about the
nature of infinity. The ancient Indians and
Greeks 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 comes
from Anaximander, a pre-Socratic Greek philosopher
who lived in Miletus. He used the word apeiron,
which means infinite or limitless. However,
the earliest attestable accounts of mathematical
infinity come from Zeno of Elea (born c. 490
BCE), a pre-Socratic Greek philosopher of
southern Italy and member of the Eleatic School
founded by Parmenides. Aristotle called him
the inventor of the dialectic. He is best
known for his paradoxes, described by Bertrand
Russell as "immeasurably subtle and profound".In
accordance with the traditional view of Aristotle,
the Hellenistic Greeks generally preferred
to distinguish the potential infinity from
the actual infinity; for example, instead
of saying that there are an infinity of primes,
Euclid prefers instead to say that there are
more prime numbers than contained in any given
collection of prime numbers.
=== Early Indian ===
The Jain mathematical text Surya Prajnapti
(c. 4th–3rd century BCE) classifies all
numbers into three sets: enumerable, innumerable,
and infinite. Each of these was further subdivided
into three orders:
Enumerable: lowest, intermediate, and highest
Innumerable: nearly innumerable, truly innumerable,
and innumerably innumerable
Infinite: nearly infinite, truly infinite,
infinitely infiniteIn this work, two basic
types of infinite numbers are distinguished.
On both physical and ontological grounds,
a distinction was made between asaṃkhyāta
("countless, innumerable") and ananta ("endless,
unlimited"), between rigidly bounded and loosely
bounded infinities.
=== 17th century ===
European mathematicians started using infinite
numbers and expressions in a systematic fashion
in the 17th century. In 1655 John Wallis first
used the notation
∞
{\displaystyle \infty }
for such a number in his De sectionibus conicis
and exploited it in area calculations by dividing
the region into infinitesimal strips of width
on the order of
1
∞
.
{\displaystyle {\tfrac {1}{\infty }}.}
But in Arithmetica infinitorum (1655 also)
he indicates infinite series, infinite products
and infinite continued fractions by writing
down a few terms or factors and then appending
"&c." For example, "1, 6, 12, 18, 24, &c."In
1699 Isaac Newton wrote about equations with
an infinite number of terms in his work De
analysi per aequationes numero terminorum
infinitas.
== Mathematics ==
Hermann Weyl opened a mathematico-philosophic
address given in 1930 with:
Mathematics is the science of the infinite.
=== Infinity symbol ===
The infinity symbol
∞
{\displaystyle \infty }
(sometimes called the lemniscate) is a mathematical
symbol representing the concept of infinity.
The symbol is encoded in Unicode at U+221E
∞ INFINITY (HTML &#8734; · &infin;) and
in LaTeX as \infty.
It was introduced in 1655 by John Wallis,
and, since its introduction, has also been
used outside mathematics in modern mysticism
and literary symbology.
=== Calculus ===
Leibniz, one of the co-inventors of infinitesimal
calculus, 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
∞
{\displaystyle \infty }
, called "infinity", is used to denote an
unbounded limit. The notation
x
→
∞
{\displaystyle x\rightarrow \infty }
means that x grows without bound, and
x
→
−
∞
{\displaystyle x\to -\infty }
means that x decreases without bound. If f(t)
≥ 0 for every t, then
∫
a
b
f
(
t
)
d
t
=
∞
{\displaystyle \int _{a}^{b}f(t)\,dt=\infty
}
means that f(t) does not bound a finite area
from
a
{\displaystyle a}
to
b
.
{\displaystyle b.}
∫
−
∞
∞
f
(
t
)
d
t
=
∞
{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=\infty
}
means that the area under f(t) is infinite.
∫
−
∞
∞
f
(
t
)
d
t
=
a
{\displaystyle \int _{-\infty }^{\infty }f(t)\,dt=a}
means that the total area under f(t) is finite,
and equals
a
.
{\displaystyle a.}
Infinity is also used to describe infinite
series:
∑
i
=
0
∞
f
(
i
)
=
a
{\displaystyle \sum _{i=0}^{\infty }f(i)=a}
means that the sum of the infinite series
converges to some real value
a
.
{\displaystyle a.}
∑
i
=
0
∞
f
(
i
)
=
∞
{\displaystyle \sum _{i=0}^{\infty }f(i)=\infty
}
means that the sum of the infinite series
diverges in the specific sense that the partial
sums grow without bound.Infinity can be used
not only to define a limit but also as a value
in the extended real number system. Points
labeled
+
∞
{\displaystyle +\infty }
and
−
∞
{\displaystyle -\infty }
can be added to the topological space of the
real numbers, producing the two-point compactification
of the real numbers. Adding algebraic properties
to this gives us the extended real numbers.
We can also treat
+
∞
{\displaystyle +\infty }
and
−
∞
{\displaystyle -\infty }
as the same, leading to the one-point compactification
of the real numbers, which is the real projective
line. Projective geometry also refers to a
line at infinity in plane geometry, a plane
at infinity in three-dimensional space, and
a hyperplane at infinity for general dimensions,
each consisting of points at infinity.
==== Complex analysis ====
In complex analysis the symbol
∞
{\displaystyle \infty }
, called "infinity", denotes an unsigned infinite
limit.
x
→
∞
{\displaystyle x\rightarrow \infty }
means that the magnitude
|
x
|
{\displaystyle |x|}
of x grows beyond any assigned value. A point
labeled
∞
{\displaystyle \infty }
can be added to the complex plane as a topological
space giving the one-point compactification
of the complex plane. When this is done, the
resulting space is a one-dimensional complex
manifold, or Riemann surface, called the extended
complex plane or the Riemann sphere. Arithmetic
operations similar to those given above for
the extended real numbers can also be defined,
though there is no distinction in the signs
(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
=
∞
{\displaystyle z/0=\infty }
for any nonzero complex number z. In this
context it is often useful to consider meromorphic
functions 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 Möbius
transformations.
=== Nonstandard analysis ===
The original formulation of infinitesimal
calculus by Isaac Newton and Gottfried 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 hyperreal
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 distinct infinite numbers. This
approach to non-standard calculus is fully
developed in Keisler (1986).
=== Set theory ===
A different form 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
aleph-null (ℵ0), 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 one-to-one 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 (however, see Galileo's
paradox 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 least one of its proper parts; this notion
of infinity is called Dedekind infinite. 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 well-ordered 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.
==== Cardinality of the continuum ====
One of Cantor's most important results was
that the cardinality of the continuum
c
{\displaystyle \mathbf {c} }
is greater than that of the natural numbers
ℵ
0
{\displaystyle {\aleph _{0}}}
; that is, there are more real numbers R than
natural numbers N. Namely, Cantor showed that
c
=
2
ℵ
0
>
ℵ
0
{\displaystyle \mathbf {c} =2^{\aleph _{0}}>{\aleph
_{0}}}
(see Cantor's diagonal argument or Cantor's
first uncountability proof).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,
c
=
ℵ
1
=
ℶ
1
{\displaystyle \mathbf {c} =\aleph _{1}=\beth
_{1}}
(see Beth one). This hypothesis can neither
be proved nor disproved within the widely
accepted Zermelo–Fraenkel 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 also that this is equal to the number
of points on a plane and, indeed, in any finite-dimensional
space.
The first of these results is apparent by
considering, for instance, the tangent function,
which provides a one-to-one correspondence
between the interval (−π/2, π/2) 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
space-filling curves, curved lines that twist
and turn enough to fill the whole of any square,
or cube, or hypercube, or finite-dimensional
space. These curves can be used to define
a one-to-one correspondence between the points
on one side of a square and the points in
the square.
=== Geometry and topology ===
Infinite-dimensional spaces are widely used
in geometry and topology, particularly as
classifying spaces, such as Eilenberg−MacLane
spaces. Common examples are the infinite-dimensional
complex projective space K(Z,2) and the infinite-dimensional
real projective space K(Z/2Z,1).
=== Fractals ===
The structure of a fractal object is reiterated
in its magnifications. Fractals can be magnified
indefinitely without losing their structure
and becoming "smooth"; they have infinite
perimeters—some with infinite, and others
with finite surface areas. One such fractal
curve with an infinite perimeter and finite
surface area is the Koch snowflake.
=== Mathematics without infinity ===
Leopold Kronecker was skeptical of the notion
of infinity and how his fellow mathematicians
were using it in the 1870s and 1880s. This
skepticism was developed in the philosophy
of mathematics called finitism, an extreme
form of mathematical philosophy in the general
philosophical and mathematical schools of
constructivism and intuitionism.
== Physics ==
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, 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 plane
wave exist, but there are no experimental
means to generate them.
=== Cosmology ===
The first published proposal that the universe
is infinite came from Thomas Digges in 1576.
Eight years later, in 1584, the Italian philosopher
and astronomer Giordano Bruno proposed an
unbounded universe in On the Infinite Universe
and Worlds: "Innumerable suns exist; innumerable
earths revolve around these suns in a manner
similar to the way the seven planets revolve
around our sun. Living beings inhabit these
worlds."Cosmologists have long sought to discover
whether infinity exists in our physical universe:
Are there an infinite number of stars? Does
the universe have infinite volume? Does space
"go on forever"? This is an open question
of cosmology. The question of being infinite
is logically separate from the question of
having boundaries. The two-dimensional surface
of the Earth, for example, is finite, yet
has no edge. By travelling in a straight line
with respect to the Earth's curvature one
will eventually return to the exact spot one
started from. The universe, at least in principle,
might have a similar topology. If so, one
might eventually return to one's starting
point after travelling in a straight line
through the universe for long enough.The curvature
of the universe can be measured through multipole
moments in the spectrum of the cosmic background
radiation. As to date, analysis of the radiation
patterns recorded by the WMAP spacecraft hints
that the universe has a flat topology. This
would be consistent with an infinite physical
universe.However, the universe could be finite,
even if its curvature is flat. An easy way
to understand this is to consider two-dimensional
examples, such as video games where items
that leave one edge of the screen reappear
on the other. The topology of such games is
toroidal and the geometry is flat. Many possible
bounded, flat possibilities also exist for
three-dimensional space.The concept of infinity
also extends to the multiverse hypothesis,
which, when explained by astrophysicists such
as Michio Kaku, posits that there are an infinite
number and variety of universes.
== Logic ==
In logic an infinite regress argument is "a
distinctively philosophical kind of argument
purporting to show that a thesis is defective
because it generates an infinite series when
either (form A) no such series exists or (form
B) were it to exist, the thesis would lack
the role (e.g., of justification) that it
is supposed to play."
== Computing ==
The IEEE floating-point standard (IEEE 754)
specifies the positive and negative infinity
values (and also indefinite values). These
are defined as the result of arithmetic overflow,
division by zero, and other exceptional operations.Some
programming languages, such as Java and J,
allow the programmer an explicit access to
the positive and negative infinity values
as language constants. These can be used as
greatest and least elements, as they compare
(respectively) greater than or less than all
other values. They have uses as sentinel 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 for a programmer to create the greatest
and least elements. In languages that do not
provide explicit access to such values from
the initial state of the program, but do implement
the floating-point data type, the infinity
values may still be accessible and usable
as the result of certain operations.In programming,
an infinite loop is a loop whose exit condition
is never satisfied, thus theoretically executing
indefinitely.
Infinite sequences can be represented in the
finite memory of a computer as a composite
data structure consisting of a few first members
of the sequence, and a recursive routine for
computing the nth element from the preceding
ones. Several techniques can be used for avoiding
computing several times the same element of
the sequence. One is lazy evaluation. Another
one, available in Maple, consists of having
routines with a remember option. This option
consists of keeping in memory the results
of the function that have been computed, and,
at each call of the routine, looking if this
particular result has been computed, for avoiding
to compute it again. For example, the standard
definition of the Fibonacci sequence is
Fib(n) == if n = 1 or n = 2 then 1 else Fib(n
- 1) + Fib(n - 2).With a standard implementation,
an exponential number of function calls is
needed for computing Fib(n), while, with the
remember option, only n - 1 function calls
are needed.
== Arts, games, and cognitive sciences ==
Perspective artwork utilizes 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 played
on an unbounded board are called infinite
chess.Cognitive scientist George Lakoff considers
the concept of infinity in mathematics and
the sciences as a metaphor. This perspective
is based on the basic metaphor of infinity
(BMI), defined as the ever-increasing sequence
.The symbol is often used romantically
to represent eternal love. Several types of
jewelry are fashioned into the infinity shape
for this purpose.
== See also ==
0.999...
Aleph number
Exponentiation
Indeterminate form
Infinite monkey theorem
Infinite set
Infinitesimal
Paradoxes of infinity
Supertask
Surreal number
== Notes
