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