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Encyclopedia > Infinity
The infinity symbol, ∞, in several typefaces.
The infinity symbol, ∞, in several typefaces.

Infinity (symbolically represented with ) comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts (usually linked to the idea of "without end") which arise in philosophy, mathematics, and theology. Infinity might refer to: Infinity, the mathematical, philosophical, or theological notion of boundlessness Infinity, a celebrity Millenium-Class cruise ship The Infinity Broadcasting Corporation, known as CBS Radio since December 14, 2005, one of the largest radio corporations in the US Infinity, a science fiction magazine Infinity, a 1996 biographical... Image File history File links Infinity_symbol. ... Image File history File links Infinity_symbol. ... For other uses, see Latins and Latin (disambiguation). ... For other uses, see Philosophy (disambiguation). ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Theology finds its scholars pursuing the understanding of and providing reasoned discourse of religion, spirituality and God or the gods. ...


In mathematics, "infinity" is often used in contexts where it is treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is a different type of "number" from the real numbers. 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. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... For other uses, see Number (disambiguation). ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, a set A is Dedekind-infinite if some proper subset B of A is equinumerous to A. Explicitly, this means that there is a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite. ... In the mathematical field of set theory, a large cardinal property is a property of cardinal numbers, such that the existence of such a cardinal provably cannot be proved within the standard axiomatic set theory ZFC, if one assumes ZFC itself is consistent. ... Part of the foundation of mathematics, Russells paradox (also known as Russells antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction. ... In Model Theory, a nonstandard model of arithmetic (or, equivalently, a nonstandard model of number theory) is a model of all of number theory (i. ... The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Newton and Leibniz. ... Projective geometry is a non-metrical form of geometry. ... The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ... The Absolute Infinite is mathematician Georg Cantors concept of an infinity that transcended the transfinite numbers. ...

Contents

History

Early Indian views of infinity

The Isha Upanishad of the Yajurveda (c. 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". The Isha Upanishad () or Ishopanishad (), also known as the Ishavasya Upanishad (), is a Sanskrit poem (or sequence of mantras) from the Upanishads and is considered Åšruti by followers of a number of diverse traditions within Hinduism. ... The Yajurveda (Sanskrit , a tatpurusha compound of sacrifice + knowledge) is one of the four Hindu Vedas. ...

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 Indian mathematical text Surya Prajnapti (c. 400 BC) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders: The Isha Upanishad () or Ishopanishad (), also known as the Ishavasya Upanishad (), is a Sanskrit poem (or sequence of mantras) from the Upanishads and is considered Åšruti by followers of a number of diverse traditions within Hinduism. ... This article is under construction. ... The Celtics claim Vienna, Austria. ...

  • 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. 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). Jain and Jaina redirect here. ... 2-dimensional renderings (ie. ...


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, ...), the smallest cardinal transfinite number. The Jains also defined a whole system of infinite cardinal numbers, of which the highest enumerable number N is the smallest. In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are generalized numbers used to measure the cardinality (size) of sets. ... Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ...


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. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In philosophy, ontology (from the Greek , genitive : of being (part. ... An asaṃkhyeya is a Buddhist name for the number 10140. ...


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."[2] An infinite regress is a series of propositions arises if the truth of proposition P1 requires the support of proposition P2, and for any proposition in the series Pn, the truth of Pn requires the support of the truth of Pn+1. ...


Infinity symbol

John Wallis introduced the infinity symbol to mathematical literature.
John Wallis introduced the infinity symbol to mathematical literature.

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." Download high resolution version (843x867, 230 KB) This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... Download high resolution version (843x867, 230 KB) This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... John Wallis John Wallis (November 22, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the development of modern calculus. ... A lemniscate In mathematics, a lemniscate is a type of curve described by a Cartesian equation of the form: Graphing this equation produces a curve similar to . ...


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. A Möbius strip made with a piece of paper and tape. ... August Ferdinand Möbius. ... Johann Benedict Listing born July 25, 1808, died December 24, 1882 was a German mathematician, born in Frankfurt, Germany, and died in Göttingen, Germany. ... Year 1858 (MDCCCLVIII) was a common year starting on Friday (link will display the full calendar) of the Gregorian Calendar (or a common year starting on Wednesday of the 12-day slower Julian calendar). ...


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. Various Religious symbols, including (first row) Christian, Jewish, Hindu, Bahai, (second row) Islamic, tribal, Taoist, Shinto (third row) Buddhist, Sikh, Hindu, Jain, (fourth row) Ayyavazhi, Triple Goddess, Maltese cross, pre-Christian Slavonic Religion is the adherence to codified beliefs and rituals that generally involve a faith in a spiritual... For other uses, see Alchemy (disambiguation). ... This article is about historical/cultural Tibet. ... For other uses, see Petroglyph (disambiguation). ... For other uses, see Ouroboros (disambiguation). ...


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.[3] John Wallis John Wallis (November 22, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the development of modern calculus. ... Events March 25 - Saturns largest moon, Titan, is discovered by Christian Huygens. ... The system of Roman numerals is a numeral system originating in ancient Rome, and was adapted from Etruscan numerals. ... The Etruscan numerals were used by the ancient Etruscans. ... Look up Ω, ω in Wiktionary, the free dictionary. ... This page contains special characters. ...


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. This article or section is in need of attention from an expert on the subject. ...


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. For other uses, see Hourglass (disambiguation). ...


The infinity symbol is represented in Unicode by the character ∞ (U+221E). The Unicode Standard, Version 5. ...


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. 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 Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... In mathematics, a series is often represented as the sum of a sequence of terms. ... It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ... Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as...

  • 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: In mathematics, a series is a sum of a sequence of terms. ...

  • sum_{i=0}^{infty} , f(i) = x means that the sum of the infinite series converges to some real value x.
  • 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 series is the sum of the terms of a sequence of numbers. ... In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a limit. ...

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. 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. 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. Projective geometry also introduces a line at infinity in plane geometry, and so forth for higher dimensions. The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, compactification is the process or result of enlarging a topological space to make it compact. ... In mathematics, compactification is the process or result of enlarging a topological space to make it compact. ... In mathematics, the projective line is a fundamental example of an algebraic curve. ... Projective geometry is a non-metrical form of geometry. ... In geometry and topology, the line at infinity is a line which is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. ... In mathematics, plane geometry may mean: geometry of the Euclidean plane; or sometimes geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others; or geometry of the hyperbolic plane or two-dimensional spherical geometry. ...


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 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 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 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. Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... 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 projective line, . Nota Bene: The real projective line is not equivalent to the extended real number line. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, compactification is the process or result of enlarging a topological space to make it compact. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... The Riemann sphere can be visualized as the complex number plane wrapped around a sphere (by some form of stereographic projection — details are given below). ... In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. ... In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ...


Nonstandard analysis

Main article: 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. In the most restricted sense, nonstandard analysis or non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of... Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... Smooth infinitesimal analysis is a mathematically rigorous reformulation of the calculus in terms of infinitesimals. ... In the most restricted sense, nonstandard analysis or non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ...


Set theory

Main articles: Cardinality and Ordinal number

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 aleph-null (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. In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ... In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ... Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are generalized numbers used to measure the cardinality (size) of sets. ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] – January 6, 1918) was a German mathematician. ... Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ... In the branch of mathematics known as set theory, aleph usually refers to a series of numbers used to represent the cardinality (or size) of infinite sets. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, IPA: ) was a German mathematician who became a logician and philosopher. ... Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...


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. 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 bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... For other uses, see Euclid (disambiguation). ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X... In set theory a set S is Dedekind-infinite if there is a bijective function from S to some proper subset of S, or equivalently if there is an injective function from the natural numbers into S. In the absence of choice, Dedekind-infinite is a stronger condition than merely...


Cantor defined two kinds of infinite numbers, the ordinal numbers and the 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. In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ... In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. ... In mathematics, a well-order (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. ... For other senses of this word, see sequence (disambiguation). ... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... In mathematics, a countable set is a set with the same cardinality (i. ... For other uses, see Number (disambiguation). ... The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Newton and Leibniz. ...


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. In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In set theory, an infinite set is a set that is not a finite set. ... Hilberts paradox of the Grand Hotel was a mathematical paradox about infinity presented by German mathematician David Hilbert (1862 – 1943): In a hotel with a finite number of rooms, it is clear that once it is full, no more guests can be accommodated. ...


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. Namely, Cantor showed that mathbf{c} = 2^{aleph_0} > {aleph_0} (see Cantor's diagonal argument). In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ... In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ... Cantors diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. ...


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). However, this hypothesis can neither be proved nor disproved within the widely accepted Zermelo-Fraenkel set theory, even assuming the Axiom of Choice. In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are generalized numbers used to measure the cardinality (size) of sets. ... In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). ... Zermelo-Fraenkel set theory, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ... 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. 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. Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ... In mathematics, the real line is simply the set of real numbers. ... The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...


The first of these results is apparent by considering, for instance, the tangent function, which provides a one-to-one correspondence between the interval [-0.5π, 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 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 in the side of a square and those in the square. In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... The term interval is used in the following contexts: cricket mathematics music time This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... Hilberts paradox of the Grand Hotel was a mathematical paradox about infinity presented by German mathematician David Hilbert (1862 – 1943): In a hotel with a finite number of rooms, it is clear that once it is full, no more guests can be accommodated. ... Giuseppe Peano Giuseppe Peano (August 27, 1858 – April 20, 1932) was an Italian mathematician and philosopher best known for his contributions to set theory. ... Space-filling curves or Peano curves are curves, first described by Giuseppe Peano, whose ranges contain the entire 2-dimensional unit square (or the 3-dimensional unit cube). ... A square A projection of a cube (into a two-dimensional image) A projection of a hypercube (into a two-dimensional image) In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ...


Cantor also showed that sets with cardinality strictly greater than mathbf c exist (see his generalized diagonal argument and theorem). They include, for instance: Cantors diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. ... In Zermelo-Fränkel set theory, Cantors theorem states that the power set (set of all subsets) of any set A has a strictly greater cardinality than that of A. Cantors theorem is obvious for finite sets, but surprisingly it holds true for infinite sets as well. ...

  • 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 {c} = beth_2 > mathbf c (see Beth two). In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). ...


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: In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ...

mathbf{c}^2 = (2^{aleph_0})^2 = 2^{2times{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 ctimesaleph_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. Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ... // 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 from natural numbers in a finite number of steps. ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ...


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[citation needed] , 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. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the word continuum sometimes denotes the real line. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics the term countable set is used to describe the size of a set, e. ... 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 operations. ... The extended real number line is obtained from the real number line R by adding two elements: +∞ and −∞ (which are not considered to be real numbers). ... The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Newton and Leibniz. ... In the physics of wave propagation (especially electromagnetic waves), a plane wave (also spelled planewave) is a constant-frequency wave whose wavefronts (surfaces of constant amplitude and phase) are infinite parallel planes normal to the propagation direction. ...


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[citation needed]. 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. The terms a priori and a posteriori are used in philosophy to distinguish between two different types of propositional knowledge. ...


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. In mathematics, a series is a sum of a sequence of terms. ... This article is about functions in mathematics. ... Quantum field theory (QFT) is the quantum theory of fields. ... Figure 1. ... 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. 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[who?] now believe the singularity may be physically real, and have since turned their attention to finding new mathematics where infinities are possible.[citation needed] This article is about the astronomical body. ... General relativity (GR) or general relativity theory (GRT) is the theory of gravitation published by Albert Einstein in 1915. ... This article or section does not cite its references or sources. ... 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 well-behaved in some particular way, such as differentiability. ...


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. 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. 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.[citation needed] For other uses, see Point of view (literature). ... This article is about the physics subject. ... For other uses, see Universe (disambiguation). ... The shape of the Universe is an informal name for a subject of investigation within physical cosmology. ... This article is about the physics subject. ... For other uses, see Topology (disambiguation). ... This box:      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 CPU and FPU implementations. ... The term arithmetic overflow or simply overflow has the following meanings. ... For the album by Hux Flux, see Division by Zero (album). ...


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). A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ... The J programming language, developed in the early 1990s by Ken Iverson and Roger Hui, is a synthesis of APL (also by Iverson) and the FP and FL functional programming languages created by John Backus (of FORTRAN, ALGOL, and BNF fame). ... The UNITY programming languages 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 is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. ... Value in mathematics refers to the quantity that is represented by a variable. ... In computer programming, a sentinel value (also referred to as a flag value, rogue value, or signal value) is a special value that is used to terminate a loop that processes structured (especially sequential) data. ... Flowcharts are often used to graphically represent algorithms. ... Sorting refers to a process of arranging items in some sequence and/or in different sets, and accordingly, it has two common, yet distinct meanings: ordering: aranging items of the same kind, class, nature, etc. ... Wikibooks has a book on the topic of How to search Look up search, searching in Wiktionary, the free dictionary. ... In signal processing, a window function (or apodization function) is a function that is zero-valued outside of some chosen interval. ... In computer programming, operator overloading (less commonly known as operator ad-hoc polymorphism) is a specific case of polymorphism in which some or all of operators like +, = or == have different implementations depending on the types of their arguments. ... In computer programming languages, a relational operator symbol or a relational operator name is a lexical or syntactic unit that denotes a relation, for example, equality or greater than, among two or more domains, the members of which are typically denoted by further expressions. ... 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 or expended to enable a particular goal. ...


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. A cube in two-point perspective. ... For other uses, see Vanishing point (disambiguation). ... 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 projective line, . Nota Bene: The real projective line is not equivalent to the extended real number line. ... Maurits Cornelis Escher (June 17, 1898 – March 27, 1972), usually referred to as M. C. Escher, was a Dutch graphic artist. ...


See also

Wikibooks
Wikibooks has a book on the topic of

Image File history File links Wikibooks-logo-en. ... Wikibooks logo Wikibooks, previously called Wikimedia Free Textbook Project and Wikimedia-Textbooks, is a wiki for the creation of books. ... In set theory, an infinite set is a set that is not a finite set. ... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... 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 of Zermelo-Fraenkel set theory. ... Hilberts paradox of the Grand Hotel was a mathematical paradox about infinity presented by German mathematician David Hilbert (1862 – 1943): In a hotel with a finite number of rooms, it is clear that once it is full, no more guests can be accommodated. ... Given enough time, a hypothetical chimpanzee typing at random would, as part of its output, almost surely produce one of Shakespeares plays (or any other text). ... The Métis Flag was first used by Métis resistance fighters in Canada prior to the Battle of Seven Oaks in 1816. ... In mathematics, the recurring decimal 0. ... In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...

Notes

  1. ^ 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).
  2. ^ Cambridge Dictionary of Philosophy, Second Edition, p. 429
  3. ^ 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.

In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ...

References

There are very few or no other articles that link to this one. ... It has been suggested that Penguin Modern Poets, Penguin Great Ideas be merged into this article or section. ... The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... Rudy Rucker, Fall 2004, photo by Georgia Rucker. ... David Foster Wallace (born February 21, 1962) is an American novelist, essayist, and short story writer, and a professor at Pomona College in Claremont, California. ...

External links


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