Infinity

For other uses, see Infinity (disambiguation).

The infinity symbol, ∞, in several typefaces.

Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. " It refers to several distinct concepts (usually linked to the idea of "without end") which arise in philosophy, mathematics, and theology. Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Theology is the study of a god or the gods from a religious perspective

In mathematics, "infinity" is often used in contexts where it is treated as if it were a number (i. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. e. , it counts or measures things: "an infinite number of terms") but it is a different type of "number" from the real numbers. In Mathematics, the real numbers may be described informally in several different ways Infinity is related to limits, aleph numbers, classes in set theory, Dedekind-infinite sets, large cardinals,^[1] Russell's paradox, non-standard arithmetic, hyperreal numbers, projective geometry, extended real numbers and the absolute Infinite. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously In Mathematics, a set A is Dedekind-infinite if some proper Subset B of A is Equinumerous to A. In the mathematical field of Set theory, a large cardinal property is a certain kind of property of Transfinite Cardinal numbers Cardinals with such properties Part of the Foundations of mathematics, Russell's paradox (also known as Russell's antinomy) discovered by Bertrand Russell in 1901 showed that the In Mathematical logic, a nonstandard model of arithmetic is a model of (first-order Peano arithmetic that contains nonstandard numbers Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced The Absolute Infinite is Mathematician Georg Cantor 's concept of an " Infinity " that transcended the Transfinite numbers Cantor

History

Early Indian views of infinity

The Isha Upanishad of the Yajurveda (c. The Isha Upanishad ( īśa upaniṣad, otherwise Ishopanishad īśopaniṣad or īśāvāsya upaniṣad) is one of the shortest of the Upanishads The Yajurveda ( Sanskrit यजुर्वेदः, a Tatpurusha compound of yajus "sacrificial formula' + veda 4th to 3rd century BC) states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity".

Pūrṇam adaḥ pūrṇam idam

Pūrṇāt pūrṇam udacyate

Pūrṇasya pūrṇam ādāya

Pūrṇam evāvasiṣyate.

That is full, this is full

From the full, the full is subtracted

When the full is taken from the full

The full still will remain — Isha Upanishad. The Isha Upanishad ( īśa upaniṣad, otherwise Ishopanishad īśopaniṣad or īśāvāsya upaniṣad) is one of the shortest of the Upanishads

The Indian mathematical text Surya Prajnapti (c. Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. 400 BC) classifies all numbers into three sets: enumerable, innumerable, and infinite. Events By place Persian Empire Artaxerxes II King of Persia appoints Tissaphernes to take over all the districts in 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

The Jains were the first to discard the idea that all infinites were the same or equal. Jainism, traditionally known as Jain Dharma / Shraman Dharma (जैन धर्म is an ancient religion of India. They recognized different types of infinities: infinite in length (one dimension), infinite in area (two dimensions), infinite in volume (three dimensions), and infinite perpetually (infinite number of dimensions). In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it

According to Singh (1987), Joseph (2000) and Agrawal (2000), the highest enumerable number N of the Jains corresponds to the modern concept of aleph-null $\aleph_0$ (the cardinal number of the infinite set of integers 1, 2, . This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. . . ), the smallest cardinal transfinite number. Transfinite numbers are Cardinal numbers or Ordinal numbers that are larger than all finite numbers yet not necessarily absolutely infinite. The Jains also defined a whole system of infinite cardinal numbers, of which the highest enumerable number N is the smallest.

In the Jaina work on the theory of sets, 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. In Philosophy, ontology (from the Greek, genitive: of being (part An asaṃkhyeya is a Buddhist name for the number 10140 The word "asaṃkhyeya" literally means "innumerable" in Sanskrit.

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. An infinite regress in a series of propositions arises if the truth of proposition P 1 requires the support of proposition P 2 and for any proposition g. , of justification) that it is supposed to play. "^[2]

Infinity symbol

John Wallis introduced the infinity symbol to mathematical literature. John Wallis ( November 23, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the

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. In Algebraic geometry, the word lemniscate refers to any of several figure-eight or ∞ shaped curves of which the best known is the Lemniscate of Bernoulli "

A popular explanation is that the infinity symbol is derived from the shape of a Möbius strip. This article is about the mathematical object See Mobius Band (music group for the music group 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. August Ferdinand Möbius ( November 17, 1790 &ndash September 26, 1868, (ˈmøbiʊs was a German Mathematician and Johann Benedict Listing ( July 25, 1808 &ndash December 24 1882) was a German Mathematician. Year 1858 ( MDCCCLVIII) was a Common year starting on Friday (link will display the full calendar of the Gregorian Calendar (or a Common

It is also possible that it is inspired by older religious/alchemical symbolism. A religion is a set of Tenets and practices often centered upon specific Supernatural and moral claims about Reality, the Cosmos Alchemy a part of the Occult Tradition is both a philosophy and a practice with an ultimately unknown aim involving the improvement of the alchemist as well as the making of "Symbolic" redirects here For other uses see Symbolism (disambiguation and Symbolic (disambiguation. For instance, it has been found in Tibetan rock carvings, and the ouroboros, or infinity snake, is often depicted in this shape. Definitions of Tibet See also Definitions of Tibet Name In English The English word Tibet, like the word for Tibet in most European Petroglyphs are Images created by removing part of a rock surface by incising pecking carving and abrading The Ouroboros (Greek grc Ουροβόρος from grc ουροβόρος όφις "tail-devouring snake" also spelled Ourorboros, Oroborus, Uroboros

John Wallis is usually credited with introducing ∞ as a symbol for infinity in 1655 in his De sectionibus conicis. John Wallis ( November 23, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the 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. Roman numerals are a Numeral system originating in ancient Rome, adapted from Etruscan numerals. The Etruscan numerals were used by the ancient Etruscans The system was adapted from the Greek Attic numerals and formed the inspiration for the later Roman " Another conjecture is that he derived it from the Greek letter ω (omega), the last letter in the Greek alphabet. OMEGA is the premier Counter-terrorism unit of Latvia. Founded in 1992 OMEGA cooperates with many other counter-terrorism units over the world The Greek alphabet (Ελληνικό αλφάβητο is a set of twenty-four letters that has been used to write the Greek language since the late 9th or early ^[3]

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. Typesetting involves the presentation of textual material in graphic form on Paper or some other medium. 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°. An hourglass, also known as a sandglass, sand timer or sand clock, is a device for the measurement of Time. 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). In Computing, Unicode is an Industry standard allowing Computers to consistently represent and manipulate text expressed in most of the world's

Mathematical infinity

Infinity is used in various branches of mathematics.

Calculus

Further information: Limit (mathematics), Series (mathematics), Improper integral

In real analysis, the symbol $\infty$ , called "infinity", denotes an unbounded limit. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with In Calculus, an improper integral is the limit of a Definite integral as an endpoint of the interval of integration approaches either a specified Real analysis is a branch of Mathematical analysis dealing with the set of Real numbers In particular it deals with the analytic properties of real In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" $x \rightarrow \infty$ 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

$\int_{a}^{b} \, f(t)\ dt \ = \infty$ means that f(t) does not bound a finite area from a to b
$\int_{-\infty}^{\infty} \, f(t)\ dt \ = \infty$ means that the area under f(t) is infinite.
$\int_{-\infty}^{\infty} \, f(t)\ dt \ = 1$ means that the area under f(t) equals 1

Infinity is also used to describe infinite series:

$\sum_{i=0}^{\infty} \, f(i) = x$ means that the sum of the infinite series converges to some real value x. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with
$\sum_{i=0}^{\infty} \, f(i) = \infty$ means that the sum of the infinite series diverges in the specific sense that the partial sums grow without bound. In Mathematics, a divergent series is an Infinite series that is not convergent, meaning that the infinite Sequence of the Partial sums

Algebraic properties

Further information: Extended real number line

Infinity is often used not only to define a limit but as a value in the affinely extended real number system. In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced Points labeled $\infty$ and $-\infty$ can be added to the topological space of the real numbers, producing the two-point compactification of the real numbers. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, compactification is the process or result of enlarging a Topological space to make it compact. Adding algebraic properties to this gives us the extended real numbers. We can also treat $\infty$ and $-\infty$ as the same, leading to the one-point compactification of the real numbers, which is the real projective line. In Mathematics, compactification is the process or result of enlarging a Topological space to make it compact. In Real analysis, the real projective line (also called the one-point compactification of the Real line, or the projectively extended real numbers Projective geometry also introduces a line at infinity in plane geometry, and so forth for higher dimensions. Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. "Ideal line" redirects here For the ideal line in racing see Racing line. In Mathematics, plane geometry may mean geometry of a plane, geometry of the Euclidean plane, or sometimes

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. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" $x \rightarrow \infty$ 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 one-point compactification of the complex plane. The point at infinity, also called ideal point, is a point which when added to the real Number line yields a Closed curve called the Real Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, compactification is the process or result of enlarging a Topological space to make it compact. 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. In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional In Mathematics, the Riemann sphere is a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that 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. In Complex analysis, a meromorphic function on an open subset D of the Complex plane is a function that is holomorphic 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. Möbius transformations should not be confused with the Möbius transform or the Möbius function.

Nonstandard analysis

Main article: Nonstandard analysis

The original formulation of the calculus by Newton and Leibniz used infinitesimal quantities. Non-standard analysis is a branch of Mathematics that formulates analysis using a rigorous notion of an Infinitesimal number 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 formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of Infinitesimals Based on the ideas of F Non-standard analysis is a branch of Mathematics that formulates analysis using a rigorous notion of an Infinitesimal number 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. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Transfinite numbers are Cardinal numbers or Ordinal numbers that are larger than all finite numbers yet not necessarily absolutely infinite. For example, if H is an infinite number, then H + H = 2H and H + 1 are different infinite numbers.

Set theory

Main articles: Cardinality and Ordinal number

A different type of "infinity" are the ordinal and cardinal infinities of set theory. In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph-null $(\aleph_0)$ , the cardinality of the set of natural numbers. Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. Transfinite numbers are Cardinal numbers or Ordinal numbers that are larger than all finite numbers yet not necessarily absolutely infinite. In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an 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. Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin &ndash 26 July 1925 Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important

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. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry 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 if some proper Subset B of A is Equinumerous to A.

Cantor defined two kinds of infinite numbers, the ordinal numbers and the cardinal numbers. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. 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. In Mathematics, a well-order relation (or well-ordering) on a set S is a Total order on S with the property that every Generalizing finite and the ordinary infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers, and transfinite sequences. In Mathematics, a sequence is an ordered list of objects (or events The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics and related technical fields the term map or mapping is often a Synonym for function. 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. A number is an Abstract object, tokens of which are Symbols used in Counting and measuring.

Our intuition gained from finite sets breaks down when dealing with infinite sets. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. In Set theory, an infinite set is a set that is not a Finite set. One example of this is Hilbert's paradox of the Grand Hotel. Hilbert's paradox of the Grand Hotel is a mathematical Paradox about Infinite sets presented by German mathematician David Hilbert (1862–1943

Cardinality of the continuum

Main article: 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 ( ${\aleph_0}$ ); that is, there are more real numbers R than natural numbers N. In Mathematics, the cardinality of the continuum, sometimes also called the power of the continuum, is the size ( Cardinality) of the set of In Mathematics, the cardinality of the continuum, sometimes also called the power of the continuum, is the size ( Cardinality) of the set of Namely, Cantor showed that $\mathbf{c} = 2^{\aleph_0} > {\aleph_0}$ (see Cantor's diagonal argument). Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891

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, $\mathbf{c} = \aleph_1 = \beth_1$ (see Beth one). In Mathematics, the continuum hypothesis (abbreviated CH) is a Hypothesis, advanced by Georg Cantor, about the possible sizes of Infinite This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. In Mathematics, the Infinite Cardinal numbers are represented by the Hebrew letter \aleph ( aleph) indexed with a subscript that runs However, this hypothesis can neither be proved nor disproved within the widely accepted Zermelo-Fraenkel set theory, even assuming the Axiom of Choice. Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory.

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 finite-dimensional space. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points 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 one-to-one correspondence between the interval [-0. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property 5π, 0. 5π] and R (see also Hilbert's paradox of the Grand Hotel). Hilbert's paradox of the Grand Hotel is a mathematical Paradox about Infinite sets presented by German mathematician David Hilbert (1862–1943 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. Giuseppe Peano ( August 27, 1858 &ndash April 20, 1932) was an Italian Mathematician, whose work was of exceptional Space-filling curves or Peano curves are Curves first described by Giuseppe Peano (1858–1932 whose ranges contain the entire 2-dimensional Unit In Geometry, a hypercube is an n -dimensional analogue of a square ( n = 2 and a Cube ( n = 3 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. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property

Cantor also showed that sets with cardinality strictly greater than $\mathbf c$ exist (see his generalized diagonal argument and theorem). Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 Note in order to fully understand this article you may want to refer to the Set theory portion of the Table of mathematical symbols. They include, for instance:

the set of all subsets of R, i. e. , the power set of R, written P(R) or 2^R
the set R^R of all functions from R to R

Both have cardinality $2^\mathbf {c} = \beth_2 > \mathbf c$ (see Beth two). In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) In Mathematics, the Infinite Cardinal numbers are represented by the Hebrew letter \aleph ( aleph) indexed with a subscript that runs

The cardinal equalities $\mathbf{c}^2 = \mathbf{c},$ $\mathbf c^{\aleph_0} = \mathbf c,$ and $\mathbf c ^{\mathbf c} = 2^{\mathbf c}$ can be demonstrated using cardinal arithmetic:

$\mathbf{c}^2 = (2^{\aleph_0})^2 = 2^{2\times{\aleph_0}} = 2^{\aleph_0} = \mathbf{c},$

$\mathbf c^{\aleph_0} = (2^{\aleph_0})^{\aleph_0} = 2^{{\aleph_0}\times{\aleph_0}} = 2^{\aleph_0} = \mathbf{c},$

$\mathbf c ^{\mathbf c} = (2^{\aleph_0})^{\mathbf c} = 2^{\mathbf c\times\aleph_0} = 2^{\mathbf c}.$

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. In Mathematics, the cardinality of the continuum, sometimes also called the power of the continuum, is the size ( Cardinality) of the set of This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued The philosophy of mathematics is the branch of Philosophy that studies the philosophical assumptions foundations and implications of Mathematics. In the Philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed In the Philosophy of mathematics

Physical infinity

In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, the word continuum has at least two distinct meanings outlined in the sections below In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an 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. In Physics, particularly in Quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical In Mathematics, the affinely extended real number system is obtained from the Real number system R by adding two elements +∞ and &minus∞ (pronounced 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. In the Physics of Wave propagation (especially Electromagnetic waves, a plane wave (also spelled planewave) is a constant-frequency wave whose

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. "A priori" redirects here For other uses see A priori. 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. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function , 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. In quantum field theory (QFT the forces between particles are mediated by other particles In Quantum field theory, the Statistical mechanics of fields and the theory of self-similar geometric structures renormalization refers to a collection One application where infinities arise is the quantification of thermodynamic temperatures. Thermodynamic temperature is the absolute measure of Temperature and is one of the principal parameters of Thermodynamics.

However, there are some currently-accepted circumstances where the end result is infinity. One example is black holes. A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e 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. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Gravitational collapse in Astronomy is the inward fall of a massive body under the influence of the force of Gravity. 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. In Mathematics, a singularity is in general a point at which a given mathematical object is not defined or a point of an exceptional set where it fails to be 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

Main article: Physical 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. Physical cosmology, as a branch of Astronomy, is the study of the large-scale structure of the Universe and is concerned with fundamental questions about its The Universe is defined as everything that Physically Exists: the entirety of Space and Time, all forms of Matter, Energy The shape of the Universe is an informal name for a subject of investigation within Physical cosmology which describes the Geometry of the Universe Physical cosmology, as a branch of Astronomy, is the study of the large-scale structure of the Universe and is concerned with fundamental questions about its Note that 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 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. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of 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. The ultimate fate of the universe is a topic in Physical cosmology.

Computer representations of infinity

The IEEE floating-point standard specifies positive and negative infinity values; these can be the result of arithmetic overflow, division by zero, or other exceptional operations. The IEEE Standard for Binary Floating-Point Arithmetic ( IEEE 754) is the most widely-used standard for floating-point computation and is followed by many The term arithmetic overflow or simply overflow has the following meanings In

Some programming languages (for example, J and UNITY) specify greatest and least elements, i. A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer. Not to be confused with the J++ or J# programming languages The J programming language, developed in the early 1990s by The UNITY programming language was constructed by K Mani Chandy and Jayadev Misra for their book Parallel Program Design A Foundation. In Mathematics, especially in Order theory, the greatest element of a subset S of a Partially ordered set (poset is an element of S e. values that compare (respectively) greater than or less than all other values. In mathematics value commonly refers to the 'output' of a function. 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 Computer programming, a sentinel value (also referred to as a flag value, rogue value, or signal value) is a special value that In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Sorting is any process of arranging items in some sequence and/or in different sets and accordingly it has two common yet distinct meanings ordering: arranging See also Window function (SQL In Signal processing, a window function (also known as an apodization function or 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). In Computer programming, operator overloading (less commonly known as operator Ad-hoc polymorphism) is a specific case of polymorphism in In Computer science a relational operator is a Programming language construct or operator that tests some kind of relation between two entities In Computer science, overhead is generally considered any combination of excess or indirect computation time memory bandwidth or other resources that are required to be utilized

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. Perspective (from Latin perspicere to see through in the graphic arts such as drawing is an approximate representation on a flat surface (such as paper of an image as it is perceived A vanishing point is a point in a perspective drawing to which Parallel lines appear to converge The point at infinity, also called ideal point, is a point which when added to the real Number line yields a Closed curve called the Real 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. Maurits Cornelis Escher (17 June 1898 – 27 March 1972 usually referred to as M

Notes

^ Large cardinals are quantitative infinities defining the number of things in a collection, which are so large that they cannot be proven to exist in the ordinary mathematics of Zermelo-Fraenkel plus Choice (ZFC). In Set theory, an infinite set is a set that is not a Finite set. Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have In Axiomatic set theory and the branches of Logic, Mathematics, and Computer science that use it the axiom of infinity is one of the Axioms Hilbert's paradox of the Grand Hotel is a mathematical Paradox about Infinite sets presented by German mathematician David Hilbert (1862–1943 The infinite monkey theorem states that a Monkey hitting keys at Random on a Typewriter keyboard for an Infinite amount of time will almost The Métis Flag was first used by Métis resistance fighters in Canada prior to the Battle of Seven Oaks in 1816 Temporal finitism is the idea that Time is Finite. The philosophy of Aristotle, expressed in such works as his Physics, held that Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common
^ Cambridge Dictionary of Philosophy, Second Edition, p. 429
^ The History of Mathematical Symbols, By Douglas Weaver, Mathematics Coordinator, Taperoo High School with the assistance of Anthony D. Smith, Computing Studies teacher, Taperoo High School.

References

Amir D. Aczel (2001). The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity. Simon & Schuster Adult Publishing Group. ISBN 0-7434-2299-6.
D. P. Agrawal (2000). D P Agrawal is an Indologist, archaeologist and author He has published works on Indian Archaeology, Metallurgy and history Ancient Jaina Mathematics: an Introduction, Infinity Foundation.
L. C. Jain (1982). Exact Sciences from Jaina Sources.
L. C. Jain (1973). "Set theory in the Jaina school of mathematics", Indian Journal of History of Science.
George G. Joseph (2000). The Crest of the Peacock: Non-European Roots of Mathematics, 2nd edition, Penguin Books. Penguin Books is a British Publisher founded in 1935 by Allen Lane. ISBN 0-14-027778-1.
Eli Maor (1991). To Infinity and Beyond. Princeton University Press. ISBN 0-691-02511-8.
John J. O'Connor and Edmund F. Robertson (1998). 'Georg Ferdinand Ludwig Philipp Cantor', MacTutor History of Mathematics archive. The MacTutor History of Mathematics archive is an award-winning website maintained by John J
John J. O'Connor and Edmund F. Robertson (2000). 'Jaina mathematics', MacTutor History of Mathematics archive. The MacTutor History of Mathematics archive is an award-winning website maintained by John J
Ian Pearce (2002). 'Jainism', MacTutor History of Mathematics archive. The MacTutor History of Mathematics archive is an award-winning website maintained by John J
Rudy Rucker (1995). Rudolf von Bitter Rucker (born March 22, 1946 in Louisville Kentucky) is an American Computer scientist and Science fiction Infinity and the Mind: The Science and Philosophy of the Infinite. Princeton University Press. ISBN 0-691-00172-3.
N. Singh (1988). 'Jaina Theory of Actual Infinity and Transfinite Numbers', Journal of Asiatic Society, Vol. 30.
David Foster Wallace (2004). David Foster Wallace (February 21 1962&ndashSeptember 12 2008 was an American author of novels, Essays and short-stories Everything and More: A Compact History of Infinity. Norton, W. W. & Company, Inc. . ISBN 0-393-32629-2.

External links

A Crash Course 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.
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) 1-59.
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

Dictionary

-noun

Limitlessness, unlimitedness, something which is growing without limits or bounds.
A number that has an infinite numerical value that cannot be counted.
A number which is very large compared to some characteristic number. For example, in optics, an object which is much further away than the focal length of a lens is said to be "at infinity", as the distance of the image from the lens varies very little as the distance increases further.
The symbol ∞.

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