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=                              Infinity                              =
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                             Introduction                             
======================================================================
Infinity represents something that is boundless or endless, or else
something that is larger than any  real or natural number. It is often
denoted by the infinity symbol .

Since the time of the ancient Greeks, 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, 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 the 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 enlarged the mathematical study of infinity by studying
infinite sets and infinite numbers, showing that they can be of
various sizes.
[https://books.google.com/books?id=LmEZMyinoecC&pg=PA616 Extract
of page 616]
For example, if a line is viewed as the set of all of its points,
their infinite number (i.e. the cardinality of the line) is larger
than the number of integers. 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, 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 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 and cosmology, whether the Universe is infinite is an open
question.


                               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 may be that of Anaximander
(c. 610 - c. 546 BC) a pre-Socratic 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 Greeks  had a "horror of the infinite"
which would, for example, explain why Euclid (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 this theorem, Euclid "was the first to
overcome the horror of the infinite". There is a similar controversy
concerning Euclid's parallel postulate, sometimes translated
:If a straight line falling across two [other] straight lines makes
internal angles on the same side [of itself whose sum is] less than
two right angles, then the two [other] straight lines, being produced
to infinity, meet on that side [of the original straight line] that
the [sum of the internal angles] is less than two right angles.
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 (c. 495 - c. 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 as "immeasurably subtle and profound".

Achilles 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 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 provided both a satisfactory
definition of a limit and a proof that, for 0 < 'x' < 1,

:'a' + 'ax' + 'ax'2 + 'ax'3 + 'ax'4 + 'ax'5 + · · · =  .

Suppose that Achilles is running at 10 meters per second, the tortoise
is walking at 0.1 meter per second, and the latter has a 100-meter
head start. The duration of the chase fits Cauchy's pattern with 'a' =
10 seconds and 'x' = 0.01.  Achilles does overtake the tortoise; it
takes him

:10 + 0.1 + 0.001 + 0.00001 + · · · =  =  = 10  seconds.


 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:

innumerable



 17th century 
==============
In the 17th century, European mathematicians started using infinite
numbers and infinite expressions in a systematic fashion. In 1655,
John Wallis 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 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 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:


 Symbol 
========
The infinity symbol  (sometimes called the lemniscate) is a
mathematical symbol representing the concept of infinity. The symbol
is encoded in Unicode at  and in LaTeX as \infty.

It was introduced in 1655 by John Wallis, and since its introduction,
it has also been used outside mathematics in modern mysticism and
literary symbology.


 Calculus 
==========
Gottfried 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 , called "infinity", is used to denote an
unbounded limit. The notation  means that '' increases without bound,
and  means that  '' decreases without bound. For example, if  for
every '', then


Infinity can also be used to describe infinite series, as follows:

value

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 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  and  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 , called "infinity", denotes an
unsigned infinite limit.  means that the magnitude  of '' grows beyond
any assigned value. A point labeled  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 (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
transformations (see Möbius transformation § Overview).


 Nonstandard analysis 
======================
The original formulation of infinitesimal calculus by Isaac Newton and
Gottfried Leibniz used infinitesimal quantities. In the 20th 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 in this sense, then H + H = 2H and H + 1 are
distinct infinite numbers. This approach to non-standard calculus is
fully developed in .


 Set theory 
============
A different form of "infinity" are the ordinal and cardinal infinities
of set theory—a system of transfinite numbers first developed by Georg
Cantor. In this system, 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 19th century from works 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 (derived from Euclid) that the whole cannot
be the same size as the part (however, see Galileo's paradox where he
concludes that positive square integers are of the same size as
positive integers). An infinite set can simply be defined as one
having the same size as at least one of its proper parts; this notion
of infinity is called Dedekind infinite. The diagram to the right
gives an example: viewing lines as infinite sets of points, the left
half of the lower blue line can be mapped in a one-to-one manner
(green correspondences) to the higher blue line, and, in turn, to the
whole lower blue line (red correspondences); therefore the whole lower
blue line and its left half have the same cardinality, i.e. "size".

Cantor defined two kinds of infinite numbers: ordinal numbers and
cardinal numbers. 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 sequences 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. 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 R than natural numbers N. Namely, Cantor
showed that  (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,   (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). 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. 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 a positive and a
negative infinity value (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.


                 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.,
[https://books.google.com/books?id=f-e0bro-0FUC&pg=PA229 Section
10-7, p. 229]
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 <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.


                               See also                               
======================================================================



 Sources 
=========

'[http://www.infinityfoundation.com/mandala/t_es/t_es_agraw_jaina.htm
Ancient Jaina Mathematics: an Introduction]',
[http://infinityfoundation.com Infinity Foundation].

philosophy. Revised 2009.

'Indian Journal of History of Science'.

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

[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cantor.html
'Georg Ferdinand Ludwig Philipp Cantor'], 'MacTutor History of
Mathematics archive'.

[http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Jaina_mathematics.html
'Jaina mathematics'], 'MacTutor History of Mathematics archive'.

[http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch5.html
'Jainism'], 'MacTutor History of Mathematics archive'.



                            External links                            
======================================================================

in the Mathematics of Infinite Sets]', 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.

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) 1-59.

[https://web.archive.org/web/20040910082530/http://www.c3.lanl.gov/mega-math/workbk/infinity/infinity.html
Hotel Infinity]

[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cantor.html
'Georg Ferdinand Ludwig Philipp Cantor'], 'MacTutor History of
Mathematics archive'.

[http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Jaina_mathematics.html
'Jaina mathematics'], 'MacTutor History of Mathematics archive'.

[http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch5.html
'Jainism'], 'MacTutor History of Mathematics archive'.

[https://www.webcitation.org/query?url=http://uk.geocities.com/frege%40btinternet.com/cantor/Phil-Infinity.htm&date=2009-10-25+04:16:26
Source page on medieval and modern writing on Infinity]

[https://www.washingtonpost.com/wp-srv/style/longterm/books/chap1/mysteryaleph.htm
The Mystery Of The Aleph: Mathematics, the Kabbalah, and the Search
for Infinity]

(compilation of articles about infinity in physics, mathematics, and
philosophy)


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=========
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License URL: http://creativecommons.org/licenses/by-sa/3.0/
Original Article: http://en.wikipedia.org/wiki/Infinity


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