1.
Infinity
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Infinity is an abstract concept describing something without any bound or larger than any number. In mathematics, infinity is treated as a number but it is not the same sort of number as natural or real numbers. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th, in the theory he developed, there are infinite sets of different sizes. For example, the set of integers is countably infinite, while the set of real numbers is uncountable. Ancient cultures had various ideas about the nature of infinity, the ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept. The earliest recorded idea of infinity comes from Anaximander, a pre-Socratic Greek philosopher who lived in Miletus and he used the word apeiron which means infinite or limitless. However, the earliest attestable accounts of mathematical infinity come from Zeno of Elea, aristotle called him the inventor of the dialectic. He is best known for his paradoxes, described by Bertrand Russell as immeasurably subtle, however, recent readings of the Archimedes Palimpsest have found that Archimedes had an understanding about actual infinite quantities. The Jain mathematical text Surya Prajnapti classifies all numbers into three sets, enumerable, innumerable, and infinite, on both physical and ontological grounds, a distinction was made between asaṃkhyāta and ananta, between rigidly bounded and loosely bounded infinities. European mathematicians started using numbers in a systematic fashion in the 17th century. John Wallis first used the notation ∞ for such a number, euler used the notation i for an infinite number, and exploited it by applying the binomial formula to the i th power, and infinite products of i factors. In 1699 Isaac Newton wrote about equations with an number of terms in his work De analysi per aequationes numero terminorum infinitas. The infinity symbol ∞ is a symbol representing the concept of infinity. The symbol is encoded in Unicode at U+221E ∞ infinity and in LaTeX as \infty and it was introduced in 1655 by John Wallis, and, since its introduction, has also been used outside mathematics in modern mysticism and literary symbology. Leibniz, one of the co-inventors of infinitesimal calculus, speculated widely about infinite numbers, in real analysis, the symbol ∞, called infinity, is used to denote an unbounded limit. X → ∞ means that x grows without bound, and x → − ∞ means the value of x is decreasing without bound. ∑ i =0 ∞ f = ∞ means that the sum of the series diverges in the specific sense that the partial sums grow without bound. Infinity can be used not only to define a limit but as a value in the real number system

2.
Aleph number
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In mathematics, and in particular set theory, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets that can be well-ordered. They are named after the symbol used to them, the Hebrew letter aleph. The cardinality of the numbers is ℵ0, the next larger cardinality is aleph-one ℵ1, then ℵ2. Continuing in this manner, it is possible to define a cardinal number ℵ α for every ordinal number α, the concept and notation are due to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities. The aleph numbers differ from the infinity commonly found in algebra, alephs measure the sizes of sets, infinity, on the other hand, is commonly defined as an extreme limit of the real number line, or an extreme point of the extended real number line. ℵ0 is the cardinality of the set of all natural numbers, the set of all finite ordinals, called ω or ω0, has cardinality ℵ0. A set has cardinality ℵ0 if and only if it is countably infinite, examples of such sets are the set of all square numbers, the set of all cubic numbers, the set of all fourth powers. These infinite ordinals, ω, ω+1, ω·2, ω2, ωω, for example, the sequence of all positive odd integers followed by all positive even integers is an ordering of the set of positive integers. If the axiom of choice holds, then ℵ0 is smaller than any other infinite cardinal. ℵ1 is the cardinality of the set of all ordinal numbers. This ω1 is itself a number larger than all countable ones. Therefore, ℵ1 is distinct from ℵ0, the definition of ℵ1 implies that no cardinal number is between ℵ0 and ℵ1. If the axiom of choice is used, it can be proved that the class of cardinal numbers is totally ordered. Using the axiom of choice we can show one of the most useful properties of the set ω1, any countable subset of ω1 has an upper bound in ω1. This fact is analogous to the situation in ℵ0, every set of natural numbers has a maximum which is also a natural number. ω1 is actually a useful concept, if somewhat exotic-sounding, an example application is closing with respect to countable operations, e. g. trying to explicitly describe the σ-algebra generated by an arbitrary collection of subsets. This is harder than most explicit descriptions of generation in algebra because in those cases we only have to close with respect to finite operations—sums, products, and the like. In popular books ℵ1 is sometimes defined to be 2 ℵ0

3.
Apeirogon
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In geometry, an apeirogon is a generalized polygon with a countably infinite number of sides. It can be considered as the limit of a polygon as n approaches infinity. The interior of an apeirogon can be defined by a direction order of vertices. This article describes an apeirogon in its form as a tessellation or partition of a line. A regular apeirogon has equal edge lengths, just like any regular polygon and its Schläfli symbol is, and its Coxeter-Dynkin diagram is. It is the first in the family of regular hypercubic honeycombs. This line may be considered as a circle of radius, by analogy with regular polygons with great number of edges. In two dimensions, a regular apeirogon divides the plane into two half-planes as a regular apeirogonal dihedron, the interior of an apeirogon can be defined by its orientation, filling one half plane. Dually the apeirogonal hosohedron has digon faces and a vertex figure. A truncated apeirogonal hosohedron becomes a apeirogonal prism, with each vertex bounded by two squares and an apeirogon, an alternated apeirogonal prism is a apeirogonal antiprism, with each vertex bounded by three triangles and an apeirogon. The regular apeirogon can also be seen as linear sets within 4 of the regular, uniform tilings, an isogonal apeirogon has a single type of vertex and alternates two types of edges. A quasiregular apeirogon is an isogonal apeirogon with equal edge lengths, an isotoxal apeirogon, being the dual of an isogonal one, has one type of edge, and two types of vertices, and is therefore geometrically identical to the regular apeirogon. It can be seen by drawing vertices in alternate colors. All of these will have half the symmetry of the regular apeirogon, Regular apeirogons that are scaled to converge at infinity have the symbol and exist on horocycles, while more generally they can exist on hypercycles. The regular tiling has regular apeirogon faces, hypercyclic apeirogons can also be isogonal or quasiregular, with truncated apeirogon faces, t, like the tiling tr, with two types of edges, alternately connecting to triangles or other apeirogons. Apeirogonal tiling Apeirogonal prism Apeirogonal antiprism Apeirohedron Circle Coxeter, H. S. M. Regular Polytopes, Regular polyhedra - old and new, Aequationes Math. 16 p. 1-20 Coxeter, H. S. M. & Moser, W. O. J. Generators, archived from the original on 4 February 2007

4.
Cantor's diagonal argument
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Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began. The diagonal argument was not Cantors first proof of the uncountability of the real numbers, diagonalization arguments are often also the source of contradictions like Russells paradox and Richards paradox. Historically, the argument first appeared in the work of Paul du Bois-Reymond in 1875. In his 1891 article, Cantor considered the set T of all sequences of binary digits. To prove this, given an enumeration of elements from T, in the example, this yields, By construction, s differs from each sn, since their nth digits differ. Hence, s cannot occur in the enumeration, based on this theorem, Cantor then uses a proof by contradiction to show that, The set T is uncountable. He assumes for contradiction that T was countable, then all its elements could be written as an enumeration s1, s2, …, sn, …. Applying the previous theorem to this enumeration would produce a sequence s not belonging to the enumeration, however, s was an element of T and should therefore be in the enumeration. This contradicts the assumption, so T must be uncountable. The interpretation of Cantors result will depend upon ones view of mathematics, to constructivists, the argument shows no more than that there is no bijection between the natural numbers and T. It does not rule out the possibility that the latter are subcountable, the uncountability of the real numbers was already established by Cantors first uncountability proof, but it also follows from the above result. To see this, we build a one-to-one correspondence between the set T of infinite binary strings and a subset of R. Since T is uncountable, this subset of R must be uncountable, to construct this one-to-one correspondence, observe that a string in T, such as t = 0111…, appears after the binary point in the binary expansion of a number between 0 and 1. This suggests defining the function f =0. t, where t is a string in T, however, this function is not injective. For instance, f =0. 1000… = 1/2, and f =0. 0111… = 1/4 + 1/8 + 1/16 + … = 1/2, map to the same number, 1/2. Modifying this function produces a bijection from T to the open interval =, the idea is to remove the problem elements from T and, and handle them separately. From, remove the numbers having two binary expansions and these are numbers of the form m/2n, where m is an odd integer and n is a natural number. Put these numbers in a sequence, a =, from T, remove the strings appearing after the binary point in the binary expansions of 0,1, and the numbers in sequence a

5.
Countable set
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In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A countable set is either a set or a countably infinite set. Some authors use countable set to mean countably infinite alone, to avoid this ambiguity, the term at most countable may be used when finite sets are included and countably infinite, enumerable, or denumerable otherwise. Georg Cantor introduced the term countable set, contrasting sets that are countable with those that are uncountable, today, countable sets form the foundation of a branch of mathematics called discrete mathematics. A set S is countable if there exists a function f from S to the natural numbers N =. If such an f can be found that is also surjective, in other words, a set is countably infinite if it has one-to-one correspondence with the natural number set, N. As noted above, this terminology is not universal, some authors use countable to mean what is here called countably infinite, and do not include finite sets. Alternative formulations of the definition in terms of a function or a surjective function can also be given. In 1874, in his first set theory article, Cantor proved that the set of numbers is uncountable. In 1878, he used one-to-one correspondences to define and compare cardinalities, in 1883, he extended the natural numbers with his infinite ordinals, and used sets of ordinals to produce an infinity of sets having different infinite cardinalities. A set is a collection of elements, and may be described in many ways, one way is simply to list all of its elements, for example, the set consisting of the integers 3,4, and 5 may be denoted. This is only effective for small sets, however, for larger sets, even in this case, however, it is still possible to list all the elements, because the set is finite. Some sets are infinite, these sets have more than n elements for any integer n, for example, the set of natural numbers, denotable by, has infinitely many elements, and we cannot use any normal number to give its size. Nonetheless, it out that infinite sets do have a well-defined notion of size. To understand what this means, we first examine what it does not mean, for example, there are infinitely many odd integers, infinitely many even integers, and infinitely many integers overall. However, it out that the number of even integers. This is because we arrange things such that for every integer, or, more generally, n→2n, see picture. However, not all sets have the same cardinality

6.
Division by zero
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In mathematics, division by zero is division where the divisor is zero. Such a division can be expressed as a/0 where a is the dividend. In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by 0, gives a, and so division by zero is undefined. Since any number multiplied by zero is zero, the expression 0/0 also has no defined value, in computing, a program error may result from an attempt to divide by zero. When division is explained at the elementary level, it is often considered as splitting a set of objects into equal parts. As an example, consider having ten cookies, and these cookies are to be distributed equally to five people at a table, each person would receive 105 =2 cookies. Similarly, if there are ten cookies, and only one person at the table, so, for dividing by zero, what is the number of cookies that each person receives when 10 cookies are evenly distributed amongst 0 people at a table. Certain words can be pinpointed in the question to highlight the problem, the problem with this question is the when. There is no way to evenly distribute 10 cookies to nobody, in mathematical jargon, a set of 10 items cannot be partitioned into 0 subsets. So 100, at least in elementary arithmetic, is said to be either meaningless, similar problems occur if one has 0 cookies and 0 people, but this time the problem is in the phrase the number. A partition is possible, but since the partition has 0 parts, vacuously every set in our partition has a number of elements, be it 0,2,5. If there are, say,5 cookies and 2 people, in any integer partition of a 5-set into 2 parts, one of the parts of the partition will have more elements than the other. But the problem with 5 cookies and 2 people can be solved by cutting one cookie in half, the problem with 5 cookies and 0 people cannot be solved in any way that preserves the meaning of divides. Another way of looking at division by zero is that division can always be checked using multiplication. Considering the 10/0 example above, setting x = 10/0, if x equals ten divided by zero, then x times zero equals ten, but there is no x that, when multiplied by zero, gives ten. If instead of x=10/0 we have x=0/0, then every x satisfies the question what number x, multiplied by zero, the Brahmasphutasiddhanta of Brahmagupta is the earliest known text to treat zero as a number in its own right and to define operations involving zero. The author could not explain division by zero in his texts, according to Brahmagupta, A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is zero or is expressed as a fraction with zero as numerator

7.
Heh (god)
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Ḥeḥ was in Egyptian mythology, the deification of infinity or eternity in the Ogdoad, his name itself meaning endlessness. His female counterpart was known as Hauhet, which is simply the feminine form of his name, like the other concepts in the Ogdoad, his male form was often depicted as a frog, or a frog-headed human, and his female form as a snake or snake-headed human. Depictions of this also had a shen ring at the base of each palm stem. Depictions of Huh were also used in hieroglyphs to represent one million, thus this deity is also known as the god of millions of years. The primary meaning of the term ḥeḥ was million or millions, subsequently, together with his female counterpart Ḥauḥet, Ḥeḥ represented a member of the Ogdoad of eight primeval deities whose worship was centred at Hermopolis Magna. The other members of the Ogdoad are Nu and Naunet, Amun and Amaunet, Kuk, the god Ḥeḥ was usually depicted anthropomorphically, as in the hieroglyphic character, as a male figure with divine beard and lappet wig. Normally kneeling, the god typically holds in each hand a palm branch. Occasionally, a palm branch is worn on the gods head. The personified, somewhat abstract god of eternity Ḥeḥ possessed no known cult centre or sanctuary, rather, his veneration revolved around symbolism, barta, Winfried, Die Bedeutung der Personifikation Huh im Unterschied zu den Personifikationen Hah und Nun, Göttinger Miszellen 127, pp. 7–12

8.
Infinite monkey theorem
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In fact the monkey would almost surely type every possible finite text an infinite number of times. One of the earliest instances of the use of the metaphor is that of French mathematician Émile Borel in 1913. Variants of the theorem include multiple and even infinitely many typists, in the early 20th century, Borel and Arthur Eddington used the theorem to illustrate the timescales implicit in the foundations of statistical mechanics. There is a proof of this theorem. As an introduction, recall that if two events are independent, then the probability of both happening equals the product of the probabilities of each one happening independently. Suppose the typewriter has 50 keys, and the word to be typed is banana, if the keys are pressed randomly and independently, it means that each key has an equal chance of being pressed. Then, the chance that the first letter typed is b is 1/50, and the chance that the letter typed is a is also 1/50. Therefore, the chance of the first six letters spelling banana is × × × × × =6 = 1/15625000000, less than one in 15 billion, but not zero, hence a possible outcome. From the above, the chance of not typing banana in a block of 6 letters is 1 −6. Because each block is typed independently, the chance Xn of not typing banana in any of the first n blocks of 6 letters is X n = n, as n grows, Xn gets smaller. For an n of a million, Xn is roughly 0.9999, but for an n of 10 billion Xn is roughly 0.53 and for an n of 100 billion it is roughly 0.0017. As n approaches infinity, the probability Xn approaches zero, that is, by making n large enough, Xn can be made as small as is desired, and the chance of typing banana approaches 100%. The same argument shows why at least one of many monkeys will produce a text as quickly as it would be produced by a perfectly accurate human typist copying it from the original. In this case Xn = n where Xn represents the probability that none of the first n monkeys types banana correctly on their first try, the limit, for n going to infinity, is zero. So the probability of the word banana appearing at some point after a number of keystrokes is equal to one. Given an infinite sequence of strings, where each character of each string is chosen uniformly at random. Both follow easily from the second Borel–Cantelli lemma, for the second theorem, let Ek be the event that the kth string begins with the given text. However, for physically meaningful numbers of monkeys typing for physically meaningful lengths of time the results are reversed, ignoring punctuation, spacing, and capitalization, a monkey typing letters uniformly at random has a chance of one in 26 of correctly typing the first letter of Hamlet

9.
Infinity mirror
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An infinity mirror is a pair of parallel mirrors, which create a series of smaller and smaller reflections that appear to recede into an infinite distance. They are used as accents and in artwork. When an outside observer looks into the surface of the reflective mirror. Alternatively, this effect can also be seen when an observer stands between two parallel fully reflective mirrors, as in some dressing rooms, some elevators, or a house of mirrors. A weaker version of this effect can be seen by standing between any two parallel surfaces, such as the glass walls of a small entry lobby into some buildings. The partially reflective glass produces this sensation, diluted by the noise of the views through the glass into the surrounding environment. The infinity mirror effect is produced there are two parallel reflective surfaces which can bounce a beam of light back and forth an indefinite number of times. The reflections appear to recede into the distance because the light actually is traversing the distance it appears to be traveling, for example, in a two-centimeter-thick infinity mirror, with the light sources halfway between, light from the source initially travels one centimeter. The first reflection travels one centimeter to the mirror and then two centimeters to, and through the front mirror, a total of three centimeters. Each successive reflection adds four more centimeters to the total, each additional reflection adds length to the path the light must travel before exiting the mirror. If the mirrors are not precisely parallel, but instead are canted at a slight angle, in theory, such a surface is infinite in area, but encloses a finite volume. Visual artists, especially contemporary sculptors, have use of infinity mirrors. Yayoi Kusama, Josiah McElheny, Ivan Navarro, and Taylor Davis have all produced works that use the infinity mirror to expand the sensation of unlimited space in their artworks. Some amusement park rides, such as Disneys Space Mountain roller coaster attraction. Corner reflector Kaleidoscope Recursion § In art

10.
Infinity symbol
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The infinity symbol ∞ is a mathematical symbol representing the concept of infinity. The shape of a figure eight has a long pedigree, for instance, it appears in the cross of Saint Boniface. However, John Wallis is credited with introducing the infinity symbol with its meaning in 1655. Leonhard Euler used a variant of the symbol in order to denote absolutus infinitus. Euler freely performed various operations on infinity, such as taking its logarithm and this symbol is not used anymore, and does not exist in Unicode. In mathematics, the infinity symbol is used often to represent a potential infinity, rather than to represent an actually infinite quantity such as the ordinal numbers. The poem after which Pale Fire is entitled explicitly refers to the miracle of the lemniscate, the well known shape and meaning of the infinity symbol have made it a common typographic element of graphic design. For instance, the Métis flag, used by the Canadian Métis people in the early 19th century, is based around this symbol. In modern commerce, corporate logos featuring this symbol have been used by, among others, Room for PlayStation Portable, Microsoft Visual Studio, Fujitsu, the symbol is encoded in Unicode at U+221E ∞ infinity and in LaTeX as \infty, ∞. The acid-free paper symbol mentioned above is encoded separately as U+267E ♾ PERMANENT PAPER SIGN

11.
Non-standard analysis
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The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals, Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson and he wrote, the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the stages of the Differential and Integral Calculus. Robinson continued, However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort, as a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits. The key to our method is provided by the analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory. In 1973, intuitionist Arend Heyting praised non-standard analysis as a model of important mathematical research. A non-zero element of an ordered field F is infinitesimal if and only if its value is smaller than any element of F of the form 1 n, for n. Ordered fields that have infinitesimal elements are also called non-Archimedean, more generally, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle. A field which satisfies the principle for real numbers is a hyperreal field. Robinsons original approach was based on these models of the field of real numbers. His classic foundational book on the subject Non-standard Analysis was published in 1966 and is still in print, on page 88, Robinson writes, The existence of non-standard models of arithmetic was discovered by Thoralf Skolem. Skolems method foreshadows the ultrapower construction Several technical issues must be addressed to develop a calculus of infinitesimals, for example, it is not enough to construct an ordered field with infinitesimals. See the article on numbers for a discussion of some of the relevant ideas. In this section we outline one of the simplest approaches to defining a hyperreal field ∗ R, let R be the field of real numbers, and let N be the semiring of natural numbers. Denote by R N the set of sequences of real numbers, a field ∗ R is defined as a suitable quotient of R N, as follows. Take a nonprincipal ultrafilter F ⊂ P, in particular, F contains the Fréchet filter. There are at least three reasons to consider non-standard analysis, historical, pedagogical, and technical, much of the earliest development of the infinitesimal calculus by Newton and Leibniz was formulated using expressions such as infinitesimal number and vanishing quantity

12.
Point at infinity
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In geometry, a point at infinity or ideal point is an idealized limiting point at the end of each line. In the case of a plane, there is one ideal point for each pencil of parallel lines of the plane. Adjoining these points produces a plane, in which no point can be distinguished. This holds for a geometry over any field, and more generally over any division ring, in the real case, a point at infinity completes a line into a topologically closed curve. In higher dimensions, all the points at infinity form a subspace of one dimension less than that of the whole projective space to which they belong. A point at infinity can also be added to the line, thereby turning it into a closed surface known as the complex projective line, CP1. In the case of a space, each line has two distinct ideal points. Here, the set of ideal points takes the form of a quadric, in an affine or Euclidean space of higher dimension, the points at infinity are the points which are added to the space to get the projective completion. As a projective space over a field is an algebraic variety. Similarly, if the field is the real or the complex field. In artistic drawing and technical perspective, the projection on the plane of the point at infinity of a class of parallel lines is called their vanishing point. In hyperbolic geometry, points at infinity are typically named ideal points, all points at infinity together form the Cayley absolute or boundary of a hyperbolic plane. This construction can be generalized to topological spaces, projective line is the Alexandroff extension of the corresponding field. Thus the circle is the one-point compactification of the line. Division by zero Midpoint § Generalizations Asymptote § Algebraic curves

13.
Projectively extended real line
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In real analysis, the projectively extended real line, is the extension of the number line by a point denoted ∞. It is thus the set R ∪ with the arithmetic operations extended where possible. The added point is called the point at infinity, because it is considered as a neighbour of both ends of the real line, more precisely, the point at infinity is the limit of every sequence of real numbers whose absolute values are increasing and unbounded. The projectively extended real line may be identified with the line over the reals in which three points have been assigned specific values. The projectively extended real line must not be confused with the real number line, in which +∞. Unlike most mathematical models of the concept of number, this structure allows division by zero. In particular 1/0 = ∞, and moreover 1/∞ =0, making reciprocal, 1/x, the structure, however, is not a field, and none of the binary arithmetic operations are total, as witnessed for example by 0⋅∞ being undefined despite the reciprocal being total. It has usable interpretations, however – for example, in geometry, the projectively extended real line extends the field of real numbers in the same way that the Riemann sphere extends the field of complex numbers, by adding a single point called conventionally ∞. In contrast, the real number line distinguishes between + ∞ and − ∞. The order relation cannot be extended to R ^ in a meaningful way, given a number a ≠ ∞, there is no convincing argument to define either a > ∞ or that a < ∞. Since ∞ cant be compared with any of the other elements, However, order on R is used in definitions in R ^. Fundamental to the idea that ∞ is a point no different from any other is the way the real line is a homogeneous space. For example the general group of 2×2 real invertible matrices has a transitive action on it. The group action may be expressed by Möbius transformations, with the understanding that when the denominator of the linear transformation is 0. This cannot be extended to 4-tuples of points, because the cross-ratio is invariant, the terminology projective line is appropriate, because the points are in 1-to-1 correspondence with one-dimensional linear subspaces of R2. The arithmetic operations on this space are an extension of the operations on reals. A motivation for the new definitions is the limits of functions of real numbers. Consequently, they are undefined, ∞ + ∞ ∞ − ∞ ∞ ⋅00 ⋅ ∞ ∞ / ∞0 /0 The following equalities mean

14.
Surreal number
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The surreals share many properties with the reals, including the usual arithmetic operations, as such, they form an ordered field. The surreals also contain all transfinite ordinal numbers, the arithmetic on them is given by the natural operations, research on the go endgame by John Horton Conway led to another definition and construction of the surreal numbers. Conways construction was introduced in Donald Knuths 1974 book Surreal Numbers, How Two Ex-Students Turned on to Pure Mathematics, in his book, which takes the form of a dialogue, Knuth coined the term surreal numbers for what Conway had called simply numbers. Conway later adopted Knuths term, and used surreals for analyzing games in his 1976 book On Numbers and Games. In the Conway construction, the numbers are constructed in stages. Different subsets may end up defining the same number, and may define the number even if L ≠ L′. So strictly speaking, the numbers are equivalence classes of representations of form that designate the same number. In the first stage of construction, there are no previously existing numbers so the representation must use the empty set. This representation, where L and R are both empty, is called 0, subsequent stages yield forms like, =1 =2 =3 and = −1 = −2 = −3 The integers are thus contained within the surreal numbers. Similarly, representations arise like, = 1/2 = 1/4 = 3/4 so that the rationals are contained within the surreal numbers. Thus the real numbers are also embedded within the surreals, but there are also representations like = ω = ε where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number. The construction consists of three interdependent parts, the rule, the comparison rule and the equivalence rule. A form is a pair of sets of numbers, called its left set. A form with left set L and right set R is written, when L and R are given as lists of elements, the braces around them are omitted. Either or both of the left and right set of a form may be the empty set, the form with both left and right set empty is also written. The numeric forms are placed in classes, each such equivalence class is a surreal number. The elements of the left and right set of a form are drawn from the universe of the surreal numbers, equivalence Rule Two numeric forms x and y are forms of the same number if and only if both x ≤ y and y ≤ x. An ordering relationship must be antisymmetric, i. e. it must have the property that x = y only when x and y are the same object and this is not the case for surreal number forms, but is true by construction for surreal numbers

15.
Temporal finitism
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Temporal finitism is the doctrine that time is finite in the past. The philosophy of Aristotle, expressed in works as his Physics, held that although space was finite, with only void existing beyond the outermost sphere of the heavens. This caused problems for mediaeval Islamic, Jewish, and Christian philosophers, modern cosmogony accepts finitism, in the form of the Big Bang, rather than Steady State theory which allows for an infinite universe, but on physical rather than philosophical grounds. This view was inspired by the creation myth shared by the three Abrahamic religions, Judaism, Christianity and Islam, prior to Maimonides, it was held that it was possible to prove, philosophically, creation theory. The Kalam cosmological argument held that creation was provable, for example, Maimonides himself held that neither creation nor Aristotles infinite time were provable, or at least that no proof was available. Thomas Aquinas was influenced by this belief, and held in his Summa Theologica that neither hypothesis was demonstrable, some of Maimonides Jewish successors, including Gersonides and Crescas, conversely held that the question was decidable, philosophically. John Philoponus was probably the first to use the argument that infinite time is impossible in order to establish temporal finitism and he was followed by many others including St. Bonaventure. Philoponus arguments for temporal finitism were severalfold, contra Aristotlem has been lost, and is chiefly known through the citations used by Simplicius of Cilicia in his commentaries on Aristotles Physics and De Caelo. A full exposition of Philoponus several arguments, as reported by Simplicius, but since Aristotle holds that such treatments of infinity are impossible and ridiculous, the world cannot have existed for infinite time. An infinite temporal regress of events is an actual infinite, thus an infinite temporal regress of events cannot exist. This argument depends on the assertion that an actual infinite cannot exist, and that an infinite past implies an infinite succession of events, a word no clearly defined. The second argument, the argument from the impossibility of completing an actual infinite by addition, states. The temporal series of past events has been completed by successive addition, thus the temporal series of past events cannot be an actual infinite. The first statement states, correctly, that a finite cannot be made into a one by the finite addition of more finite numbers. The second skirts around this, the idea in mathematics, that the sequence of negative integers. -3, -2, -1 may be extended by appending zero, then one. Now the infinity of a series consists in the fact that it can never be completed through successive synthesis. It thus follows that it is impossible for an infinite world-series to have passed away, for most purposes it is simply used as convenient, when considered more carefully it is incorporated, or not, according to whether the axiom of infinity is included. This is the concept of infinity, while this may provide useful analogies or ways of thinking about the physical world

16.
John Wallis
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John Wallis was an English mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later and he is credited with introducing the symbol ∞ for infinity. He similarly used 1/∞ for an infinitesimal, Wallis was born in Ashford, Kent, the third of five children of Reverend John Wallis and Joanna Chapman. He was initially educated at a school in Ashford but moved to James Movats school in Tenterden in 1625 following an outbreak of plague, as it was intended that he should be a doctor, he was sent in 1632 to Emmanuel College, Cambridge. While there, he kept an act on the doctrine of the circulation of the blood and his interests, however, centred on mathematics. He received his Bachelor of Arts degree in 1637 and a Masters in 1640, from 1643 to 1649, he served as a nonvoting scribe at the Westminster Assembly. He was elected to a fellowship at Queens College, Cambridge in 1644, throughout this time, Wallis had been close to the Parliamentarian party, perhaps as a result of his exposure to Holbeach at Felsted School. He rendered them great practical assistance in deciphering Royalist dispatches, most ciphers were ad hoc methods relying on a secret algorithm, as opposed to systems based on a variable key. Wallis realised that the latter were far more secure – even describing them as unbreakable and he was also concerned about the use of ciphers by foreign powers, refusing, for example, Gottfried Leibnizs request of 1697 to teach Hanoverian students about cryptography. Returning to London – he had been chaplain at St Gabriel Fenchurch in 1643 – Wallis joined the group of scientists that was later to evolve into the Royal Society. He was finally able to indulge his interests, mastering William Oughtreds Clavis Mathematicae in a few weeks in 1647. He soon began to write his own treatises, dealing with a range of topics. Wallis joined the moderate Presbyterians in signing the remonstrance against the execution of Charles I, in spite of their opposition he was appointed in 1649 to the Savilian Chair of Geometry at Oxford University, where he lived until his death on 28 October 1703. In 1661, he was one of twelve Presbyterian representatives at the Savoy Conference, besides his mathematical works he wrote on theology, logic, English grammar and philosophy, and he was involved in devising a system for teaching deaf mutes. William Holder had earlier taught a man, Alexander Popham, to speak plainly and distinctly. Wallis later claimed credit for this, leading Holder to accuse Wallis of rifling his Neighbours, Wallis made significant contributions to trigonometry, calculus, geometry, and the analysis of infinite series. In his Opera Mathematica I he introduced the term continued fraction, Wallis rejected as absurd the now usual idea of a negative number as being less than nothing, but accepted the view that it is something greater than infinity. In 1655, Wallis published a treatise on conic sections in which they were defined analytically and this was the earliest book in which these curves are considered and defined as curves of the second degree

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Where Mathematics Comes From
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Where Mathematics Comes From, How the Embodied Mind Brings Mathematics into Being is a book by George Lakoff, a cognitive linguist, and Rafael E. Núñez, a psychologist. Published in 2000, WMCF seeks to found a science of mathematics. Nikolay Lobachevsky said There is no branch of mathematics, however abstract, a common type of conceptual blending process would seem to apply to the entire mathematical procession. Lakoff and Núñezs avowed purpose is to begin laying the foundations for a scientific understanding of mathematics. They find that four distinct but related processes metaphorically structure basic arithmetic, object collection, object construction, using a measuring stick, WMCF builds on earlier books by Lakoff and Lakoff and Johnson, which analyze such concepts of metaphor and image schemata from second-generation cognitive science. Some of the concepts in these books, such as the interesting technical ideas in Lakoff, are absent from WMCF. Lakoff and Núñez hold that mathematics results from the human cognitive apparatus, Lakoff and Núñez start by reviewing the psychological literature, concluding that human beings appear to have an innate ability, called subitizing, to count, add, and subtract up to about 4 or 5. They document this conclusion by reviewing the literature, published in recent decades, for example, infants quickly become excited or curious when presented with impossible situations, such as having three toys appear when only two were initially present. The authors argue that mathematics goes far beyond this very elementary level due to a number of metaphorical constructions. Much of WMCF deals with the important concepts of infinity and of limit processes, thus much of WMCF is, in effect, a study of the epistemological foundations of the calculus. Lakoff and Núñez conclude that while the infinite is not metaphorical. Moreover, they deem all manifestations of actual infinity to be instances of what they call the Basic Metaphor of Infinity, WMCF emphatically rejects the Platonistic philosophy of mathematics. They emphasize that all we know and can know is human mathematics. The question of there is a transcendent mathematics independent of human thought is a meaningless question. That is like asking if colors are transcendent of human thought- colors are only varying wavelengths of light, WMCF likewise criticizes the emphasis mathematicians place on the concept of closure. Lakoff and Núñez argue that the expectation of closure is an artifact of the minds ability to relate fundamentally different concepts via metaphor. WMCF concerns itself mainly with proposing and establishing a view of mathematics, one grounding the field in the realities of human biology. It is not a work of technical mathematics or philosophy, Lakoff and Núñez are not the first to argue that conventional approaches to the philosophy of mathematics are flawed