Epsilon Numbers (mathematics)

	Epsilon Numbers (mathematics) In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplication. The original epsilon numbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers ''ε'' that satisfy the equation :\varepsilon = \omega^\varepsilon, \, in which ω is the smallest infinite ordinal. The least such ordinal is ''ε''0 (pronounced epsilon nought or epsilon zero), which can be viewed as the "limit" obtained by transfinite recursion from a sequence of smaller limit ordinals: :\varepsilon_0 = \omega^ = \sup \\,, where is the supremum function, which is equivalent to set union in the case of the von Neumann representation of ordinals. Larger ordinal fixed points of the exponential map are indexed by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Fixed-point Lemma For Normal Functions The fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed points (Levy 1979: p. 117). It was first proved by Oswald Veblen in 1908. Background and formal statement A normal function is a class function f from the class Ord of ordinal numbers to itself such that: * f is strictly increasing: f(\alpha) whenever $\alpha<\beta$ . * $f$ is continuous: for every limit ordinal $\lambda$ (i.e. $\lambda$ is neither zero nor a successor), $f(\lambda)=\sup\$ . It can be shown that if $f$ is normal then $f$ commutes with suprema; for any nonempty set $A$ of ordinals, : $f(\sup A)=\sup f(A) = \sup\$ . Indeed, if $\sup A$ is a successor ordinal then $\sup A$ is an element of $A$ an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Normal Function In axiomatic set theory, a function ''f'' : Ord → Ord is called normal (or a normal function) if and only if it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two conditions: # For every limit ordinal ''γ'' (i.e. ''γ'' is neither zero nor a successor), it is the case that ''f''(''γ'') = sup . # For all ordinals ''α'' < ''β'', it is the case that ''f''(''α'') < ''f''(''β''). Examples A simple normal function is given by (see ordinal arithmetic ). But is ''not'' normal because it is not continuous at any limit ordinal; that is, the inverse image of the one-point open set is the set , which is not open when ''λ'' is a limit ordinal. If ''β'' is a fixed ordinal, then the functions , (for ), and (for ) are all normal. More important exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Map (mathematics) In mathematics, a map or mapping is a function in its general sense. These terms may have originated as from the process of making a geographical map: ''mapping'' the Earth surface to a sheet of paper. The term ''map'' may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning or it may mean a linear polynomial. In category theory, a map may refer to a morphism. The term ''transformation'' can be used interchangeably, but ''transformation'' often refers to a function from a set to itself. There are also a few less common uses in logic and graph theory. Maps as functions In many branches of mathematics, the term ''map'' is used to mean a function, sometimes with a specific property of particular importance to that branch. For instance, a "map" is a " continuous function" in topology, a "linear transformation" in linear algebra, etc. Some ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Limit Ordinal In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists an ordinal γ such that β 0, are limits of limits, etc. Properties The classes of successor ordinals and limit ordinals (of various cofinalities) as well as zero exhaust the entire class of ordinals, so these cases are often used in proofs by transfinite induction or definitions by transfinite recursion. Limit ordinals represent a sort of "turning point" in such procedures, in which one must use limiting operations such as taking the union over all preceding ordinals. In principle, one could do anything at limit ordinals, but taking the union is continuous in the order topology and this is usually desirable. If we use the von Neumann cardinal assignment, every infinite cardinal number is also a limit ordinal (and this is a fitting obs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Ordinal Exponentiation In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the result of the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. In addition to these usual ordinal operations, there are also the "natural" arithmetic of ordinals and the nimber operations. Addition The union of two disjoint well-ordered sets ''S'' and ''T'' can be well-ordered. The order-type of that union is the ordinal that results from adding the order-types of ''S'' and ''T''. If two well-ordered sets are not already disjoint, then they can be replaced by order-isomorphic disjoint sets, e.g. replace ''S'' by × ''S'' and ''T'' by × ''T''. This way, the well-ordered set ''S'' is written "to the left" of the well-ordere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Additively Indecomposable Ordinal In set theory, a branch of mathematics, an additively indecomposable ordinal ''α'' is any ordinal number that is not 0 such that for any \beta,\gamma<\alpha, we have $\beta+\gamma<\alpha.$ Additively indecomposable ordinals are also called ''gamma numbers'' or ''additive principal numbers''. The additively indecomposable ordinals are precisely those ordinals of the form $\omega^\beta$ for some ordinal $\beta$ . From the continuity of addition in its right argument, we get that if $\beta < \alpha$ and ''α'' is additively indecomposable, then $\beta + \alpha = \alpha.$ Obviously 1 is additively indecomposable, since $0+0<1.$ No finite ordinal other than $1$ is additively indecomposable. Also, $\omega$ is additively indecomposable, since the sum of two finit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Additively Indecomposable Ordinal In set theory, a branch of mathematics, an additively indecomposable ordinal ''α'' is any ordinal number that is not 0 such that for any \beta,\gamma<\alpha, we have $\beta+\gamma<\alpha.$ Additively indecomposable ordinals are also called ''gamma numbers'' or ''additive principal numbers''. The additively indecomposable ordinals are precisely those ordinals of the form $\omega^\beta$ for some ordinal $\beta$ . From the continuity of addition in its right argument, we get that if $\beta < \alpha$ and ''α'' is additively indecomposable, then $\beta + \alpha = \alpha.$ Obviously 1 is additively indecomposable, since $0+0<1.$ No finite ordinal other than $1$ is additively indecomposable. Also, $\omega$ is additively indecomposable, since the sum of two finit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Surreal Number In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers (including the hyperreal numbers) can be realized as subfields of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown (in von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Donald Knuth Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer science. Knuth has been called the "father of the analysis of algorithms". He is the author of the multi-volume work ''The Art of Computer Programming'' and contributed to the development of the rigorous analysis of the computational complexity of algorithms and systematized formal mathematical techniques for it. In the process, he also popularized the asymptotic notation. In addition to fundamental contributions in several branches of theoretical computer science, Knuth is the creator of the TeX computer typesetting system, the related METAFONT font definition language and rendering system, and the Computer Modern family of typefaces. As a writer and scholar, Knuth created the WEB and CWEB computer programming systems designed to encou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. On 11 April 2020, at age 82, he died of complications from COVID-19. Early life and education Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius College, Camb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]

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