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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 ... Cardinal numbers References {{settheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinal Exponentiation
In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case of infinite sets, the infinite cardinal numbers have been introduced, which are often denoted with the Hebrew letter \aleph (aleph) marked with subscript indicating their rank among the infinite cardinals. 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 number of elements. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for two infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Pure And Applied Logic
The ''Annals of Pure and Applied Logic'' is a peer-reviewed scientific journal published by Elsevier that publishes papers on applications of mathematical logic in mathematics, in computer science, and in other related disciplines.About Annals of Pure and Applied Logic The editors of ''Annals of Pure and Applied Logic'' include mathematicians Ulrich Kohlenbach at TU Darmstadt in Germany, Thomas Scanlon at |
Cardinal Number
In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the case of infinite sets, the infinite cardinal numbers have been introduced, which are often denoted with the Hebrew letter \aleph (aleph) marked with subscript indicating their rank among the infinite cardinals. 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 number of elements. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for two infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is gre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of is variously denoted as , , , \mathbb(S), or . Any subset of is called a ''family of sets'' over . Example If is the set , then all the subsets of are * (also denoted \varnothing or \empty, the empty set or the null set) * * * * * * * and hence the power set of is . Properties If is a finite set with the cardinality (i.e., the number of all elements in the set is ), then the number of all the subsets of is . This fact as well as the reason of the notation denoting the power set are demonstrated in the below. : An indicator function or a characteristic function of a subset of a set with the cardinality is a function from to the two-element set , denoted as , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuum Hypothesis
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: 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 '' continuum'' for the real numbers. History Cantor believed the continuum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality Of The Continuum
In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \bold\mathfrak c (lowercase Fraktur "c") or \bold, \bold\mathbb R\bold, . The real numbers \mathbb R are more numerous than the natural numbers \mathbb N. Moreover, \mathbb R has the same number of elements as the power set of \mathbb N. Symbolically, if the cardinality of \mathbb N is denoted as \aleph_0, the cardinality of the continuum is This was proven by Georg Cantor in his uncountability proof of 1874, part of his groundbreaking study of different infinities. The inequality was later stated more simply in his diagonal argument in 1891. Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between them. Between any two real numbers ''a'' < ''b'', no matter how close they ... [...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 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 or the gimel function \gimel. Definition Beth numbers are defined by transfinite recursion: * \beth_0 = \aleph_0, * \beth_ = 2^, * \beth_\lambda = \sup\Bigl\, 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_\a ... [...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 of ''α'') and : Statement If is a class function whose domain consists of ordinals and whose range consists of ordinals such that # is non-decreasing, # th ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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. 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 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 value allowed by the axioms of Zermelo–Fraenkel set theory (assuming consistency). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
