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Epsilon Number
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 (chiefly British), epsilon naught (chiefly American), 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 \left\\,, where is the supremum, which is equivalent to set union in the case of the von Neumann representation of ordinals. Larger ordinal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peano Arithmetic
In mathematical logic, the Peano axioms (, ), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. The axiomatization of arithmetic provided by Peano axioms is commonly called Peano arithmetic. The importance of formalizing arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Map (mathematics)
In mathematics, a map or mapping is a function (mathematics), function in its general sense. These terms may have originated as from the process of making a map, 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 (function), 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 (mathematics), function, sometimes with a specific property of particular importance to that branch. For instance, a "map" is a "continuous f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 In mathematics, a cardinal number, or ... [...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 sum of two well-ordered sets and is the ordinal representing the variant of lexicographical order with least significant position first, on the union of the Cartesian products and . This way, every element of is smaller than every element of , comparisons within keep the order they already have, and likewise for comparisons within . The definition of addition can also be given by transfinite recursion on . When the right a ... [...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 Additively indecomposable ordinals were named the ''gamma numbers'' by Cantor,A. Rhea, The Ordinals as a Consummate Abstraction of Number Systems (2017), preprint.p.20 and are also called ''additive principal numbers''. The class (set theory), class of additively indecomposable ordinals may be denoted , from the German "Hauptzahl".W. Pohlers, "A short course in ordinal analysis", pp. 27–78. Appearing in Peter Aczel, Aczel, Simmons, ''Proof Theory: A selection of papers from the Leeds Proof Theory Programme 1990'' (1992). Cambridge University Press, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surreal Number
In mathematics, the surreal number system is a total order, totally ordered proper class containing not only the real numbers but also Infinity, infinite and infinitesimal, infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. Research on the Go endgame by John Horton Conway led to the original definition and construction of surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book ''Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness''. 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 numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donald Knuth
Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist and mathematician. He is a 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". Knuth is the author of the multi-volume work '' The Art of Computer Programming''. He 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 des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was 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 Coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Veblen Function
In mathematics, the Veblen functions are a hierarchy of normal functions ( continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If ''φ''0 is any normal function, then for any non-zero ordinal ''α'', ''φ''''α'' is the function enumerating the common fixed points of ''φ''''β'' for ''β''<''α''. These functions are all normal. Veblen hierarchy In the special case when ''φ''0(''α'')=ω''α'' this family of functions is known as the Veblen hierarchy. The function ''φ''1 is the same as the ε function: ''φ''1(''α'')= ε''α''. If then .M. Rathjen[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Analysis
In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength. If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory. In addition to obtaining the proof-theoretic ordinal of a theory, in practice ordinal analysis usually also yields various other pieces of information about the theory being analyzed, for example characterizations of the classes of provably recursive, hyperarithmetical, or \Delta^1_2 functions of the theory. History The field of ordinal analysis was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof-theoretic ordinal of Peano arithmetic is ε0. See Gentzen's consistency proof. Definition Ordinal analysis concerns true, effective (recursive) theories that can interpret a sufficient ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
