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Transfinite Cardinal Numbers
In mathematics, transfinite numbers or infinite numbers are numbers that are " infinite" in the sense that they are larger than all finite numbers. 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 in 1895 by Georg Cantor, 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. Notable work on transfinite numbers was done by Wacław Sierpiński: ''Leçons sur les nombres transfinis'' (1928 book) much expanded into '' Cardinal and Ordinal Numbers'' (1958, 2nd ed. 1965). Definition Any finite natural ... [...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] |
Aleph-one
In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ). The smallest cardinality of an infinite set is that of the natural numbers, denoted by \aleph_0 (read ''aleph-nought'', ''aleph-zero'', or ''aleph-null''); the next larger cardinality of a well-ordered set is \aleph_1, then \aleph_2, then \aleph_3, and so on. Continuing in this manner, it is possible to define an infinite cardinal number \aleph_ for every ordinal number \alpha, as described below. 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 (\infty) commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Actual Infinity
In the philosophy of mathematics, the abstraction of actual infinity, also called completed infinity, involves infinite entities as given, actual and completed objects. The concept of actual infinity was introduced into mathematics near the end of the 19th century by Georg Cantor with his theory of infinite sets, and was later formalized into Zermelo–Fraenkel set theory. This theory, which is presently commonly accepted as a foundation of mathematics, contains the axiom of infinity, which means that the natural numbers form a set (necessarily infinite). A great discovery of Cantor is that, if one accepts infinite sets, then there are different sizes ( cardinalities) of infinite sets, and, in particular, the cardinal of the continuum of the real numbers is strictly larger than the cardinal of the natural numbers. Actual infinity is to be contrasted with potential infinity, in which an endless process (such as "add 1 to the previous number") produces a sequence with no last el ... [...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 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 ... [...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 f ... [...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] |
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] |
Hyperreal Number
In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite if, and only if, , x, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom Of Countable Choice
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function A with domain \mathbb (where \mathbb denotes the set of natural numbers) such that A(n) is a non-empty set for every n\in\mathbb, there exists a function f with domain \mathbb such that f(n)\in A(n) for every n\in\mathbb. Applications ACω is particularly useful for the development of mathematical analysis, where many results depend on having a choice function for a countable collection of sets of real numbers. For instance, in order to prove that every accumulation point x of a set S\subseteq\mathbb is the limit of some sequence of elements of S\setminus\, one needs (a weak form of) the axiom of countable choice. When formulated for accumulation points of arbitrary metric spaces, the statement becomes equivalent to ACω. The ability to perform analysis using ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind-infinite Set
In mathematics, a set ''A'' is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset ''B'' of ''A'' is equinumerous to ''A''. Explicitly, this means that there exists a bijective function from ''A'' onto some proper subset ''B'' of ''A''. A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection exists). Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers. A simple example is \mathbb, the set of natural numbers. From Galileo's paradox, there exists a bijection that maps every natural number ''n'' to its square ''n''2. Since the set of squares is a proper subset of \mathbb, \mathbb is Dedekind-infinite. Until the foundational crisis of mathematics showed the need for a more careful treatment of set theory, most mathematicians assumed that a set is infinite if and only if it is Dedekind-infinite. In the early twentieth ... [...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 fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
