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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] |
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] |
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] |
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 : The sum here is the cardinality of the of the sets ''mi'', and the product is the cardinality of the . 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] |
