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Transfinite Cardinal Numbers
In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets. The term ''transfinite'' was coined by Georg Cantor in 1895, who wished to avoid some of the implications of the word ''infinite'' in connection with these objects, which were, nevertheless, not ''finite''. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as infinite numbers. Nevertheless, the term "transfinite" also remains in use. Definition Any finite natural number can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (e.g., a bag of five marbles), whereas ordinal numbers specify th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Zermelo–Fraenkel Set Theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models from containing u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean E
Jean may refer to: People * Jean (female given name) * Jean (male given name) * Jean (surname) Fictional characters * Jean Grey, a Marvel Comics character * Jean Valjean, fictional character in novel ''Les Misérables'' and its adaptations * Jean Pierre Polnareff, a fictional character from ''JoJo's Bizarre Adventure'' Places * Jean, Nevada, USA; a town * Jean, Oregon, USA Entertainment * Jean (dog), a female collie in silent films * "Jean" (song) (1969), by Rod McKuen, also recorded by Oliver * ''Jean Seberg'' (musical), a 1983 musical by Marvin Hamlisch Other uses * JEAN (programming language) * USS ''Jean'' (ID-1308), American cargo ship c. 1918 * Sternwheeler Jean, a 1938 paddleboat of the Willamette River See also * Jehan * * Gene (other) * Jeanne (other) * Jehanne (other) * Jeans (other) Jeans are denim trousers. Jeans may also refer to: Astronomy * Jeans (lunar crater) * Jeans (Martian crater) * 2763 Jeans, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MacTutor History Of Mathematics Archive
The MacTutor History of Mathematics archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland. It contains detailed biographies on many historical and contemporary mathematicians, as well as information on famous curves and various topics in the history of mathematics. The History of Mathematics archive was an outgrowth of Mathematical MacTutor system, a HyperCard database by the same authors, which won them the European Academic Software award in 1994. In the same year, they founded their web site. it has biographies on over 2800 mathematicians and scientists. In 2015, O'Connor and Robertson won the Hirst Prize of the London Mathematical Society for their work... The citation for the Hirst Prize calls the archive "the most widely used and influential web-based resource in history of mathematics". See also * Mathematics Genealogy Project * MathWorld * PlanetMath PlanetMath is a free, collaborative, m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " infinity- th" item in a sequence. Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are the reciprocals of one another. Infinitesimal numbers were introduced in the development of calculus, in which the derivative was first conceived as a ratio of two infinitesimal quantities. This definition was not rigorously formalized. As calculus developed further, infinitesimals were replaced by limits, which can be calculated using the standard real numbers. Infinitesimals regained popularit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Beth Number
In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written \beth_0,\ \beth_1,\ \beth_2,\ \beth_3,\ \dots, where \beth is the second Hebrew letter ( beth). The beth numbers are related to the aleph numbers (\aleph_0,\ \aleph_1,\ \dots), but unless the generalized continuum hypothesis is true, there are numbers indexed by \aleph that are not indexed by \beth. Definition Beth numbers are defined by transfinite recursion: * \beth_0=\aleph_0, * \beth_=2^, * \beth_=\sup\, where \alpha is an ordinal and \lambda is a limit ordinal. The cardinal \beth_0=\aleph_0 is the cardinality of any countably infinite set such as the set \mathbb of natural numbers, so that \beth_0=, \mathbb, . Let \alpha be an ordinal, and A_\alpha be a set with cardinality \beth_\alpha=, A_\alpha, . Then, *\mathcal(A_\alpha) denotes the power set of A_\alpha (i.e., the set of all subsets of A_\alpha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Actual Infinity
In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These might include the set of natural numbers, extended real numbers, transfinite numbers, or even an infinite sequence of rational numbers. Actual infinity is to be contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps. As a result, potential infinity is often formalized using the concept of Limit (mathematics), limit. Anaximander The ancient Greek term for the potential or improper infinite was ''Apeiron (cosmology), apeiron'' (unlimited or indefinite), in contrast to the actual or proper infinite ''aphorismenon''. ''Apeiron'' stands opposed to that which has a ''peras'' (limit). These notions are tod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Arithmetic
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-ordered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
On Numbers And Games
''On Numbers and Games'' is a mathematics book by John Horton Conway first published in 1976. The book is written by a pre-eminent mathematician, and is directed at other mathematicians. The material is, however, developed in a playful and unpretentious manner and many chapters are accessible to non-mathematicians. Martin Gardner discussed the book at length, particularly Conway's construction of surreal numbers, in his Mathematical Games column in ''Scientific American'' in September 1976. The book is roughly divided into two sections: the first half (or ''Zeroth Part''), on numbers, the second half (or ''First Part''), on games. In the ''Zeroth Part'', Conway provides axioms for arithmetic: addition, subtraction, multiplication, division and inequality. This allows an axiomatic construction of numbers and ordinal arithmetic, namely, the integers, reals, the countable infinity, and entire towers of infinite ordinals. The object to which these axioms apply takes the form , whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
