Cardinality of the continuumIn set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers , sometimes called the continuum. It is an infinite cardinal number and is denoted by (lowercase Fraktur "c") or . The real numbers are more numerous than the natural numbers . Moreover, has the same number of elements as the power set of Symbolically, if the cardinality of is denoted as , 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.
Limit cardinalIn mathematics, limit cardinals are certain cardinal numbers. A cardinal number λ is a weak limit cardinal if λ is neither a successor cardinal nor zero. This means that one cannot "reach" λ from another cardinal by repeated successor operations. These cardinals are sometimes called simply "limit cardinals" when the context is clear. A cardinal λ is a strong limit cardinal if λ cannot be reached by repeated powerset operations. This means that λ is nonzero and, for all κ < λ, 2κ < λ.
Club setIn mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal that is closed under the order topology, and is unbounded (see below) relative to the limit ordinal. The name club is a contraction of "closed and unbounded". Formally, if is a limit ordinal, then a set is closed in if and only if for every if then Thus, if the limit of some sequence from is less than then the limit is also in If is a limit ordinal and then is unbounded in if for any there is some such that If a set is both closed and unbounded, then it is a club set.
Cofinal (mathematics)In mathematics, a subset of a preordered set is said to be cofinal or frequent in if for every it is possible to find an element in that is "larger than " (explicitly, "larger than " means ). Cofinal subsets are very important in the theory of directed sets and nets, where “cofinal subnet” is the appropriate generalization of "subsequence".
Easton's theoremIn 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 2κ when κ is a regular cardinal are (where cf(α) is the cofinality of α) and If G is a class function whose domain consists of ordinals and whose range consists of ordinals such that G is non-decreasing, the cofinality of is greater than for each α in the domain of G, and is regular for each α in the domain of G, then there is a model of ZFC such that for each in the domain of G.
CofinalityIn 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 . This second definition makes sense without the axiom of choice.
Forcing (mathematics)In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe to a larger universe by introducing a new "generic" object . Forcing was first used by Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory.
Constructible universeIn mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by , is a particular class of sets that can be described entirely in terms of simpler sets. is the union of the constructible hierarchy . It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis".
Aleph numberIn 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 (). The cardinality of the natural numbers is (read aleph-nought or aleph-zero; the term aleph-null is also sometimes used), the next larger cardinality of a well-ordered set is aleph-one then and so on.
Inaccessible cardinalIn set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal κ is strongly inaccessible if it is uncountable, it is not a sum of fewer than κ cardinals smaller than κ, and implies . The term "inaccessible cardinal" is ambiguous. Until about 1950, it meant "weakly inaccessible cardinal", but since then it usually means "strongly inaccessible cardinal".
Ordinal numberIn set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element").
