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Encyclopedia > Infinite sets

Infinity refers to several distinct concepts which arise in theology, philosophy, mathematics and everyday life. Popular or colloquial usage of the term often does not accord with its more technical meanings. The word infinity comes from the Latin infinitas, "unboundedness". Theology is reasoned discourse concerning God (Greek θεος, theos, God, + λογος, logos, word or reason). It can also refer to the study of other religious topics. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Latin is an ancient Indo-European language originally spoken in the region around Rome called Latium. ...


In theology, for example in the work of theologians such as Duns Scotus, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity. In philosophy, infinity can be attributed to space and time, as for instance in Kant's first antinomy. In both theology and philosophy, infinity is explored in articles such as the Ultimate, the Absolute, God, and Zeno's paradoxes. Theology is reasoned discourse concerning God (Greek θεος, theos, God, + λογος, logos, word or reason). It can also refer to the study of other religious topics. ... This is a list of notable Christian theologians. ... Blessed John Duns Scotus (c. ... Michelangelos depiction of God in the painting Creation of the Sun and Moon in the Sistine Chapel) This article discusses the term God in the context of monotheism and derived henotheistic forms. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... It has been suggested that Kantianism be merged into this article or section. ... Antinomy (Greek anti-, against, plus nomos, law) is a term used in logic and epistemology, which, loosely, means a paradox or unresolvable contradiction. ... The Ultimate is a general term embracing the concept of an ultimate supernatural reality which transcends material reality and from which, according to a broad spectrum of Eastern philosophies and religions, material reality derives. ... The Absolute is the totality of things, all that is, whether it has been discovered or not. ... Zenos paradoxes are a set of paradoxes devised by Zeno of Elea to support Parmenides doctrine that all is one and that contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion. ...


In mathematics, infinity is relevant to, or the subject matter of, articles such as mathematical limits, aleph numbers, classes in set theory, Dedekind-infinite sets, large cardinals, Russell's paradox, hyperreal numbers, projective geometry, extended real numbers and the Absolute Infinite. By some, infinity is considered to be not a number but a concept of increase beyond bounds. 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 larger and larger; or the behavior of a sequences elements, as their index becomes larger and larger. ... In the branch of mathematics known as set theory, the aleph numbers are a series 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 mathematics, a cardinal is called a large cardinal if it belongs to a class of cardinals, the existence of which provably cannot be proved within the standard axiomatic set theory ZFC, if one assumes ZFC itself is consistent. ... Russells paradox (also known as Russells antinomy) is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Frege is contradictory. ... In mathematics, particularly in non-standard analysis and mathematical logic, hyperreal numbers or nonstandard reals (usually denoted as *R) denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. ... Projective geometry can be thought of informally as the geometry which arises from placing ones eye at a point. ... 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 Georg Cantors concept of an infinity that transcended the transfinite numbers. ...


In popular culture, we have Buzz Lightyear's rallying cry, "To infinity — and beyond!", which may also be viewed as the rallying cry of set theorists considering large cardinals.1 Popular culture, or pop culture, is the vernacular (peoples) culture that prevails in any given society. ... Spoiler warning: Buzz Lightyear is a fictional character appearing in several CGI films and cartoons by Disney and Pixar. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, a cardinal is called a large cardinal if it belongs to a class of cardinals, the existence of which provably cannot be proved within the standard axiomatic set theory ZFC, if one assumes ZFC itself is consistent. ...


For a discussion about infinity and the physical universe, see Universe. The deepest visible-light image of the cosmos. ...

Contents


History

Ancient view of infinity

The earliest known documented knowledge of infinity is presented in the Veda- Yajur Veda which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". The Indian Jaina mathematical text Surya Prajinapti (ca. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. It recognises five different types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually. [1] [2] The concept of different orders of infinity would remain unknown in Europe until the late 19th century. The Vedas are part of the Hindu Shruti; these religious scriptures form part of the core of the Brahminical and Vedic traditions within Hinduism and are the inspirational, metaphysical and mythological foundation for later Vedanta, Yoga, Tantra and even Bhakti forms of Hinduism. ... The Yajur Veda यजुर्वेद is one of the four Hindu Vedas; it contains religious texts focussing on liturgy and ritual. ... JAIN is an activity within the Java Community Process, developing APIs for the creation of telephony (voice and data) services. ... Centuries: 5th century BC - 4th century BC - 3rd century BC Decades: 450s BC 440s BC 430s BC 420s BC 410s BC - 400s BC - 390s BC 380s BC 370s BC 360s BC 350s BC Years: 405 BC 404 BC 403 BC 402 BC 401 BC - 400 BC - 399 BC 398 BC... For other uses, see Europe (disambiguation). ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ...


In Europe, the traditional view derives from Aristotle: Aristotle, marble copy of bronze by Lysippos. ...

"... it is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. Hence the infinite is potential, never actual; the number of parts that can be taken always surpasses any assigned number." [Physics 207b8]

This is often called potential infinity; however there are two ideas mixed up with this. One is that it is always possible to find a number of things that surpasses any given number, even if there are not actually such things. The other is that we may quantify over infinite sets without restriction. For example, ∀n∈Z(∃m∈Z[m>n∧P(m)]), which reads, "for any integer n, there exists an integer m > n such that P(m)". The second view is found in a clearer form by medieval writers such as William of Ockham: In science, a magnitude is the numerical size of something: see orders of magnitude. ... For the numerical analysis algorithm, see bisection method. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... Hello, I am Sam, Sam I am. ...

"Sed omne continuum est actualiter existens. Igitur quaelibet pars sua est vere existens in rerum natura. Sed partes continui sunt infinitae quia non tot quin plures, igitur partes infinitae sunt actualiter existentes." (But every continuum is actually existent. Therefore any of its parts is really existent in nature. But the parts of the continuum are infinite because there are not so many that there are not more, and therefore the infinite parts are actually existent.)

The parts are actually there, in some sense. However, on this view, no infinite magnitude can have a number, for whatever number we can imagine, there is always a larger one: "There are not so many (in number) that there are no more". Aquinas also argued against the idea that infinity could be in any sense complete, or a totality. In mathematics, the word continuum sometimes denotes the real line. ... Saint Thomas Aquinas [Thomas of Aquin, or Aquino] (c. ...


Views from the Renaissance to modern times

Galileo (during his long house arrest in Siena after his condemnation by the Inquisition) was the first to notice that we can place an infinite set into one-to-one correspondence with one of its proper subsets (any part of the set, that is not the whole). For example, we can match up the "set" of even numbers {2, 4, 6, 8 ...} with the natural numbers {1, 2, 3, 4 ...} as follows: Galileo Galilei Galileo Galileii (Pisa, February 15, 1564 – Arcetri, January 8, 1642), was an Italian physicist, astronomer, and philosopher who is closely associated with the scientific revolution. ... This page is about Siena, Italy. ... Artistic (i. ... 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. ... 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...

1, 2, 3, 4, ...
2, 4, 6, 8, ...

It appeared, by this reasoning, as though a set which is naturally smaller than the set of which it is a part (since it does not contain all the members of that set) is in some sense the same size. He thought this was one of the difficulties which arise when we try, "with our finite minds", to comprehend the infinite.

"So far as I see we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the attributes "equal", "greater", and "less", are not applicable to infinite, but only to finite, quantities." [On two New Sciences, 1638]

The idea that size can be measured by one-to-one correspondence is today known as Hume's principle, although Hume, like Galileo, believed the principle could not be applied to infinite sets. Humes principle is a standard for comparing any two sets of objects as to size. ...


Locke, in common with most of the empiricist philosophers, also believed that we can have no proper idea of the infinite. They believed all our ideas were derived from sense data or "impressions", and since all sensory impressions are inherently finite, so too are our thoughts and ideas. Our idea of infinity is merely negative or privative. John Locke (August 29, 1632–October 28, 1704) was an influencial English philosopher and social contract theorist. ... Empiricism is generally regarded as being at the heart of the modern scientific method, that our theories should be based on our observations of the world rather than on intuition or faith; that is, empirical research and a posteriori inductive reasoning rather than purely deductive logic. ... The concept of sense data (singular: sense datum) is very influential and widely used in the philosophy of perception. ...

"Whatever positive ideas we have in our minds of any space, duration, or number, let them be never so great, they are still finite; but when we suppose an inexhaustible remainder, from which we remove all bounds, and wherein we allow the mind an endless progression of thought, without ever completing the idea, there we have our idea of infinity ... yet when we would frame in our minds the idea of an infinite space or duration, that idea is very obscure and confused, because it is made up of two parts very different, if not inconsistent. For let a man frame in his mind an idea of any space or number, as great as he will, it is plain the mind rests and terminates in that idea; which is contrary to the idea of infinity, which consists in a supposed endless progression." (Essay, II. xvii. 7., author's emphasis)

Famously, the ultra-empiricist Hobbes tried to defend the idea of a potential infinity in the light of the discovery by Evangelista Torricelli, of a figure (Gabriel's horn) whose surface area is infinite, but whose volume is finite. Not reported, this motivation of Hobbes came too late as curves having infinite length yet bounding finite areas were known much before. Such seeming paradoxes are resolved by taking any finite figure and stretching its content infinitely in one direction; the magnitude of its content is unchanged as its divisions drop off geometrically but the magnitude of its bounds increases to infinity by necessity. Potentiality lies in the definitions of this operation, as well-defined and interconsistent mathematical axioms. A potential infinity is allowed by letting an infinitely-large quantity be cancelled out by an infinitely-small quantity. Thomas Hobbes (April 5, 1588–December 4, 1679) was an English philosopher, whose famous 1651 book Leviathan set the agenda for nearly all subsequent Western political philosophy. ... Evangelista Torricelli, portrait by an unknown artist Evangelista Torricelli (October 15, 1608 - October 25, 1647) was an Italian physicist and mathematician. ... Gabriels Horn (also called Torricellis trumpet) is a figure invented by Evangelista Torricelli which has infinite surface area, but finite volume. ... This article explains the meaning of area as a Physical quantity. ... Volume, also called capacity, is a quantification of how much space an object occupies. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ...


Modern philosophical views

Modern discussion of the infinite is now regarded as part of set theory and mathematics, and generally avoided by philosophers. An exception was Wittgenstein, who made an impassioned attack upon axiomatic set theory, and upon the idea of the actual infinite, during his "middle period".2 Ludwig Josef Johann Wittgenstein (IPA: ) (April 26, 1889 – April 29, 1951) was an Austrian philosopher who contributed several ground-breaking works to modern philosophy, primarily on the foundations of logic, the philosophy of mathematics, the philosophy of language, and the philosophy of mind. ... Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. ...

"Does the relation m = 2n correlate the class of all numbers with one of its subclasses? No. It correlates any arbitrary number with another, and in that way we arrive at infinitely many pairs of classes, of which one is correlated with the other, but which are never related as class and subclass. Neither is this infinite process itself in some sense or other such a pair of classes ... In the superstition that m = 2n correlates a class with its subclass, we merely have yet another case of ambiguous grammar." (Philosophical Remarks § 141, cf Philosophical Grammar p. 465)

Unlike the traditional empiricists, he thought that the infinite was in some way given to sense experience.

"... I can see in space the possibility of any finite experience ... we recognise [the] essential infinity of space in its smallest part." "[Time] is infinite in the same sense as the three-dimensional space of sight and movement is infinite, even if in fact I can only see as far as the walls of my room."
"... what is infinite about endlessness is only the endlessness itself."

2-dimensional renderings (ie. ...

Infinity symbol

The precise origins of the infinity symbol ∞ are unclear. One possibility is suggested by the name it is sometimes called — the lemniscate, from the Latin lemniscus, meaning "ribbon". One can imagine walking forever along a simple loop formed from a ribbon. 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 improbable, 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 (November 17, 1790, Schulpforta, Saxony, Germany - September 26, 1868, Leipzig) was a German mathematician and theoretical astronomer. ... Johann Benedict Listing (born July 25, 1808, died December 24, 1882) was a German mathematician. ... 1858 is a common year starting on Friday. ...


John Wallis is usually credited with introducing ∞ as a symbol for infinity in 1655 in his De sectionibus conicus. 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. 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 May 10 - English troops land on Jamaica 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. ... Omega (Ω ω) is the 24th and last letter of the Greek alphabet. ... Note: This article contains special characters. ...


The infinity symbol is represented in Unicode by the character ∞ (∞). Technical note: Due to technical limitations, some web browsers may not display some special characters in this article. ...


Mathematical infinity

Infinity in real analysis

In real analysis, the symbol infty, called "infinity", denotes an unbounded limit. x rightarrow infty means that x grows beyond any assigned value, and x rightarrow -infty means x is eventually less than any assigned value. Points labeled infty and can be added to the real numbers as a topological space, 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 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. Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... A limit can be: Limit (mathematics), including: Limit of a function Limit of a sequence Limit point Net (topology) Limit (category theory) A constraint (mathematical, physical, economical, legal, etc. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, compactification is applied to topological spaces to make them compact spaces. ... 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). ... In mathematics, the projective line is a fundamental example of an algebraic curve. ... Projective geometry can be thought of informally as the geometry which arises from placing ones eye at a point. ... 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, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ...


Infinity is often used not only to define a limit but as if it were a value in the extended real numbers in real analysis; if f(t) ≥ 0 then Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ...

  • int_{0}^{1} , f(t) dt  = infty means that f(t) does not bound a finite area from 0 to 1
  • int_{0}^{infty} , f(t) dt  = infty means that the area under f(t) is not finite
  • int_{0}^{infty} , f(t) dt  = 1 means that the area under f(t) approaches 1

Infinity in 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 still a one-dimensional complex manifold and called the extended complex plane or the Riemann sphere. 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. Complex analysis is the branch of mathematics investigating functions of complex numbers. ... A limit can be: Limit (mathematics), including: Limit of a function Limit of a sequence Limit point Net (topology) Limit (category theory) A constraint (mathematical, physical, economical, legal, etc. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ... 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 fore the function. ... In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i. ...


Arithmetic properties of infinity

Infinity is not a real number but may be considered part of the extended real number line, in which arithmetic operations involving infinity may be performed. 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). ...


Infinity with itself

  1. infty + infty = infty cdot infty = (-infty) cdot (-infty) = infty
  2. (-infty) + (-infty) = infty cdot (-infty) = (-infty) cdot infty = -infty

Operations involving infinity and real numbers

  1. -infty < x < infty
  2. x + infty = infty and
  3. x - infty = -infty
  4. x - (-infty) = infty
  5. {x over infty} = 0 and {x over -infty} = 0
  6. If 0<x<infty then x cdot infty = infty and x cdot (-infty) = (-infty).
  7. If -infty<x<0 then x cdot infty = -infty and x cdot (-infty) = infty.

Undefined operations

  1. 0 cdot infty and 0 cdot (-infty)
  2. and

Notice that . This is because zero times infinity is undefined.


Infinity in set theory

A different type of "infinity" are the ordinal and cardinal infinities of set theory. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is 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. 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. Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ... Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ... Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ... In the branch of mathematics known as set theory, the aleph numbers are 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, injections, and surjections, and another which uses cardinal numbers. ... In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory. ... Friedrich Ludwig Gottlob Frege (8 November 1848, Wismar – 26 July 1925, Bad Kleinen) was a German mathematician who evolved into a logician and philosopher. ... Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician and Ernst Eduard Kummers closest follower in arithmetic. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... 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. ... Euclid Euclid of Alexandria (Greek: ) (ca. ... 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. Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In the branch of mathematics known as set theory, the aleph numbers are a series 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. ... This is a page about mathematics. ... 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. ... Number is the current mathematics collaboration of the week! Please help improve it to featured article standard. ... In mathematics, particularly in non-standard analysis and mathematical logic, hyperreal numbers or nonstandard reals (usually denoted as *R) denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. ...


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. Hilberts paradox of the Grand Hotel - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ...


Mathematics without infinity

Leopold Kronecker rejected the notion of infinity and began a school of thought, in the philosophy of mathematics called finitism, which led to 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 that branch of philosophy which attempts to answer questions such as: why is mathematics useful in describing nature?, in which sense(s), if any, do mathematical entities such as numbers exist? and why and how are mathematical statements true?. Various approaches to answering these questions will... 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. ...


Use of infinity in common speech

In common parlance, infinity is often used in a hyperbolic sense. For example, "The movie was infinitely boring, but we had to wait forever to get tickets." A hyperbole, largely synonymous with exaggeration and overconsulting, is a figure of speech in which statements are exaggerated or extravagant. ...


In video games, infinite lives and infinite ammo refer to a never-ending supply of lives and ammunition. An infinite loop in computer programming is a conditional loop construction whose condition always evaluates to true. In theory, as long as there is no external interaction, the loop will continue to run for all time. In practice however, some programming loops considered as infinite will halt by exceeding the (finite) number range of one of its variables. See halting problem. These terms describe things that are only potential infinities; it is impossible to play a video game for an infinite period of time or keep a computer running for an infinite period of time. This article is about computer and video games. ... Boxes of ammunition clog a warehouse in Baghdad Ammunition is a generic military term meaning (the assembly of) a projectile and its propellant. ... For example see: Infinite Loop This article is about the programming term. ... Computer code (HTML with JavaScript) in a tool that uses colors to help the developer see the function of each piece of code. ... In computability theory the halting problem is a decision problem which can be informally stated as follows: Given a description of a program and its initial input, determine whether the program, when executed on this input, ever halts (completes). ...


The number Infinity plus 1 is also used sometimes in common speech. In popular culture, infinity plus 1 is a phrase used in relation to the notion of infinity as the largest possible number. ...


Physical infinity

In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e. counting). It is therefore assumed by physicists that no measurable quantity could have an infinite value, for instance by taking an infinite value in an extended real number system (see also: hyperreal number), or by requiring the counting of an infinite number of events. It is for example presumed impossible for any body to have infinite mass or infinite energy. There exists the concept of infinite entities (such as an infinite plane wave) but there are no means to generate such things. Likewise, perpetual motion machines theoretically generate infinite energy by attaining 100% efficiency or greater, and emulate every conceivable open system; the impossible problem follows of knowing that the output is actually infinite when the source or mechanism exceeds any known and understood system. A black hole concept drawing by NASA. Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... Look up Continuum in Wiktionary, the free dictionary. ... In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory. ... The word discrete comes from the Latin word discretus which means separate. ... 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). ... In mathematics, particularly in non-standard analysis and mathematical logic, hyperreal numbers or nonstandard reals (usually denoted as *R) denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. ... 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. ... This article or section should include material from Parallel Path See also Perpetuum mobile as a musical term Perpetual motion machines (the Latin term perpetuum mobile is not uncommon) are a class of hypothetical machines which would produce useful energy in a way science cannot explain (yet). ... An open system may refer to more than one thing: In the physical sciences, an open system (system theory) is a system where matter or energy can flow into and/or out of, in contrast to a closed system, where no energy or matter may enter or leave. ... A system is an assemblage of inter-related elements comprising a unified whole. ...


This point of view does not mean that infinity cannot be used in physics. For convenience 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. In mathematics, a series is a sum of a sequence of terms. ... Partial plot of a function f. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... Figure 1. ...


Infinity in cosmology

An intriguing question is whether actual infinity exists in our physical universe: Are there infinitely many stars? Does the universe have infinite volume? Does space "go on forever"? This is an important open question of cosmology. Note that the question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By walking/sailing/driving 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. The deepest visible-light image of the cosmos. ... Cosmology, from the Greek: κοσμολογία (cosmologia, κόσμος (cosmos) world + λογια (logia) discourse) is the study of the universe in its totality and by extension mans place in it. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...


If the universe is indeed ever expanding as science suggests then you could never get back to your starting point even on an infinite time scale.


Three types of infinities

Besides the mathematical infinity and the physical infinity, there could also be a philosophical infinity. There are scientists who hold that all three really exist and there are scientists who hold that none of the three exist. And in between there are the various possibilities. Rudy Rucker, in his book Infinity and the Mind -- the science and philosophy of the mind (1982), has worked out a model list of representatives of each of the eight possible standpoints. The footnote on p.335 of his book suggests the consideration of the following names: Abraham Robinson, Plato, Thomas Aquinas, L.E.J. Brouwer, David Hilbert, Bertrand Russell, Kurt Gödel and Georg Cantor. Rudy von Bitter Rucker (born March 22, 1946) is an American computer scientist and science fiction author, often included in lists of cyberpunk authors. ... Abraham Robinson Abraham Robinson (October 6, 1918 - April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics. ... Plato Plato (Greek: Πλάτων, Plátōn) (c. ... Saint Thomas Aquinas [Thomas of Aquin, or Aquino] (c. ... Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, and measure theory and complex analysis. ... David Hilbert David Hilbert (January 23, 1862–February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... The Right Honourable Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970), was an influential British logician, philosopher, and mathematician, working mostly in the 20th century. ... Kurt Gödel Kurt Gödel [kurt gøːdl], (April 28, 1906–January 14, 1978) was a logician, mathematician, and philosopher of mathematics. ... Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845, St. ...


Infinity in science fiction

The Hitchhiker's Guide to the Galaxy contains the following definition of infinity: The cover of the first novel in the Hitchhikers series, from a late 1990s printing. ...

"Bigger than the biggest thing ever and then some, much bigger than that, in fact really amazingly immense, a totally stunning size, real 'Wow, that's big!' time. Infinity is just so big that by comparison, bigness itself looks really titchy. Gigantic multiplied by colossal multiplied by staggeringly huge is the sort of concept we are trying to get across here."

Another quote from The Hitchhiker's Guide to the Galaxy states: "Infinity itself looks flat and uninteresting. Looking up into the night sky is looking into infinity -- distance is incomprehensible and therefore meaningless." The cover of the first novel in the Hitchhikers series, from a late 1990s printing. ...


Rudy Rucker's novel White Light describes a mathematician who leaves his body and travels to a kind of afterworld that includes a mountain whose Absolute Infinite height matches that of the class of all ordinals. Georg Cantor makes an appearance as a character, and the hero finds a physical correlate for Cantor's Continuum Problem. Rudy von Bitter Rucker (born March 22, 1946) is an American computer scientist and science fiction author, often included in lists of cyberpunk authors. ...


See also

In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ... 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. ...

References

  • Aczel, Amir D. (2001). The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity, Simon & Schuster Adult Publishing Group. ISBN 0743422996.
  • Wallace, David Foster (2004). Everything and More: A Compact History of Infinity, Norton, W. W. & Company, Inc.. ISBN 0393326292.

External links

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 See also Logic of antinomies. URL accessed on November 14, 2005.

  Results from FactBites:
 
Infinity - Wikipedia, the free encyclopedia (3563 words)
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.
Generalizing finite and the ordinary infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers, and transfinite sequences.
Likewise, perpetual motion machines theoretically generate infinite energy by attaining 100% efficiency or greater, and emulate every conceivable open system; the impossible problem follows of knowing that the output is actually infinite when the source or mechanism exceeds any known and understood system.
Dedekind-infinite set - Wikipedia, the free encyclopedia (821 words)
This definition of "infinite set" should be compared and contrasted to the usual definition: a set A is finite if A is empty, or if there is a positive integer n such that A is equinumerous to the set {1, 2, 3,..., n}.
Since every infinite, well-ordered set is Dedekind-infinite, and since the AC is equivalent to the well-ordering theorem stating that every set can be well-ordered, clearly the general AC implies that every infinite set is Dedekind-infinite.
It is notable that this definition was the first definition of "infinite" which did not rely on the definition of the natural numbers.
  More results at FactBites »

 
 

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