Gimel Function

	Gimel Function In axiomatic set theory, the gimel function is the following function mapping cardinal numbers to cardinal numbers: :\gimel\colon\kappa\mapsto\kappa^ where cf denotes the cofinality function; the gimel function is used for studying the continuum function and the cardinal number#Cardinal exponentiation, cardinal exponentiation function. The symbol \gimel is a serif form of the Hebrew letter gimel. Values of the gimel function The gimel function has the property \gimel(\kappa)>\kappa for all infinite cardinals \kappa by König's theorem (set theory), König's theorem. For regular cardinals \kappa, \gimel(\kappa)= 2^\kappa, and Easton's theorem says we don't know much about the values of this function. For singular \kappa, upper bounds for \gimel(\kappa) can be found from Saharon Shelah, Shelah's PCF theory. The gimel hypothesis The gimel hypothesis states that \gimel(\kappa)=\max(2^,\kappa^+). In essence, this means that \gimel(\kappa) for singular \kappa is the smallest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Axiomatic Set Theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of ''naive set theory''. After the discovery of Paradoxes of set theory, paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set theory is co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Cardinal Number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ''transfinite'' cardinal numbers, often denoted using the Hebrew symbol \aleph ( aleph) followed by a subscript, describe the sizes of infinite sets. Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Cofinality In mathematics, especially in order theory, the cofinality cf(''A'') of a partially ordered set ''A'' is the least of the cardinalities of the cofinal subsets of ''A''. This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member. The cofinality of a partially ordered set ''A'' can alternatively be defined as the least ordinal ''x'' such that there is a function from ''x'' to ''A'' with cofinal image. This second definition makes sense without the axiom of choice. If the axiom of choice is assumed, as will be the case in the rest of this article, then the two definitions are equivalent. Cofinality can be similarly defined for a directed set and is used to generalize the notion of a subsequence in a net. Examples * The cofinality of a partially ordered set with greatest element is 1 as the set consisting only of the greatest element is cofinal (and must be contained in every other cofinal subse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Continuum Function In mathematics, the continuum function is \kappa\mapsto 2^\kappa, i.e. raising 2 to the power of κ using cardinal exponentiation. Given a cardinal number, it is the cardinality of the power set of a set of the given cardinality. See also Continuum hypothesis Cardinality of the continuum Beth number Easton's theorem Gimel function In axiomatic set theory, the gimel function is the following function mapping cardinal numbers to cardinal numbers: :\gimel\colon\kappa\mapsto\kappa^ where cf denotes the cofinality function; the gimel function is used for studying the continuum f ... Cardinal numbers {{settheory-stub ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
picture info	Gimel Gimel is the third letter of the Semitic abjads, including Phoenician Gīml , Hebrew Gimel , Aramaic Gāmal , Syriac Gāmal , and Arabic (in alphabetical order; fifth in spelling order). Its sound value in the original Phoenician and in all derived alphabets, except Arabic, is a voiced velar plosive ; in Modern Standard Arabic, it represents either a or for most Arabic speakers except in Northern Egypt, the southern parts of Yemen and some parts of Oman where it is pronounced as the voiced velar plosive ( see below). In its Proto-Canaanite form, the letter may have been named after a weapon that was either a staff sling or a throwing stick (spear thrower), ultimately deriving from a Proto-Sinaitic glyph based on the hieroglyph below: T14 The Phoenician letter gave rise to the Greek gamma (Γ), the Latin C, G, Ɣ and yogh , and the Cyrillic Г and Ґ. Hebrew gimel Variations Hebrew spelling: Bertrand Russell posits that the letter's form is a conventionali ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	König's Theorem (set Theory) In set theory, König's theorem states that if the axiom of choice holds, ''I'' is a set, \kappa_i and \lambda_i are cardinal numbers for every ''i'' in ''I'', and \kappa_i < \lambda_i for every ''i'' in ''I'', then : $\sum_\kappa_i < \prod_\lambda_i.$ The sum here is the cardinality of the disjoint union of the sets ''m_i'', and the product is the cardinality of the Cartesian product . However, without the use of the axiom of choice, the sum and the product cannot be defined as cardinal numbers, and the meaning of the inequality sign would need to be clarified. König's theorem was introduced by in the slightly weaker form that the sum of a strictly increasing sequence of nonzero cardinal numbers is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Easton's Theorem In set theory, Easton's theorem is a result on the possible cardinal numbers of powersets. (extending a result of Robert M. Solovay) showed via forcing that the only constraints on permissible values for 2''κ'' when ''κ'' is a regular cardinal are : \kappa < \operatorname(2^\kappa) (where cf(''α'') is the cofinality of ''α'') and : $\text \kappa < \lambda \text 2^\kappa\le 2^\lambda.$ Statement If ''G'' is a class function whose domain consists of ordinals and whose range consists of ordinals such that # ''G'' is non-decreasing, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Saharon Shelah Saharon Shelah ( he, שהרן שלח; born July 3, 1945) is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey. Biography Shelah was born in Jerusalem on July 3, 1945. He is the son of the Israeli poet and political activist Yonatan Ratosh. He received his PhD for his work on stable theories in 1969 from the Hebrew University. Shelah is married to Yael, and has three children. His brother, magistrate judge Hamman Shelah was murdered along with his wife and daughter by an Egyptian soldier in the Ras Burqa massacre in 1985. Shelah planned to be a scientist while at primary school, but initially was attracted to physics and biology, not mathematics. Later he found mathematical beauty in studying geometry: He said, "But when I reached the ninth grade I began studying geometry and my eyes opened to that beauty—a system of demonstration and theorems based on a very small number of axioms which impr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	PCF Theory PCF theory is the name of a mathematical theory, introduced by Saharon , that deals with the cofinality of the ultraproducts of ordered sets. It gives strong upper bounds on the cardinalities of power sets of singular cardinals, and has many more applications as well. The abbreviation "PCF" stands for "possible cofinalities". Main definitions If ''A'' is an infinite set of regular cardinals, ''D'' is an ultrafilter on ''A'', then we let \operatorname(\prod A/D) denote the cofinality of the ordered set of functions \prod A where the ordering is defined as follows: f if $\\in D$ . pcf(''A'') is the set of cofinalities that occur if we consider all ultrafilters on ''A'', that is, $\operatorname(A)=\.$ Main results Obviously, pcf(''A'') consists of regular cardinals. Considering ultrafilters concentrated on elements of ''A'', we get that $A\subseteq \operatorname(A)$ . Shelah proved, ... [...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]
	Continuum Hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: 2^=\aleph_1, or even shorter with beth numbers: \beth_1 = \aleph_1. The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940. The name of the hypothesis comes from the term '' the continuum'' for the real numbers. History Cantor believed the continuum hypothesis to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Aleph Number In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (\,\aleph\,). The cardinality of the natural numbers is \,\aleph_0\, (read ''aleph-nought'' or ''aleph-zero''; the term ''aleph-null'' is also sometimes used), the next larger cardinality of a well-orderable set is aleph-one \,\aleph_1\;, then \,\aleph_2\, and so on. Continuing in this manner, it is possible to define a cardinal number \,\aleph_\alpha\, 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 m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]

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