
Review of Short Phrases and Links 
This Review contains major "Cardinality" related terms, short phrases and links grouped together in the form of Encyclopedia article.
Definitions
 Cardinality is a measure of the size of a set.
 Cardinality is also an area studied for its own sake as part of set theory, particularly in trying to describe the properties of large cardinal s.
 A cardinality is a text string with values chosen from the set .
 The cardinality is a simple number.
 Cardinality is based on possibility for "1to1 mapping".
 With Choice, all sets have cardinality, and the cardinal numbers can be well ordered.
 In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality.
 There are two approaches to cardinality  one which compares sets directly using bijections and injections, and another which uses cardinal numbers.
 Any wellordered set having that ordinal as its order type has the same cardinality.
 That is, there is a Dedekind infinite set A such that the cardinality of A is m.
 Without the axiom of choice, there might exist cardinalities incomparable to alephnull (such as the cardinality of a Dedekindfinite infinite set).
 Sets A and B have the same cardinality if there is a onetoone function f with domain A and range B. We denote this by  A  =  B . Definition.
 This motivates the definition of an infinite set being any set which has a proper subset of the same cardinality; in this case is a proper subset of .
 The alephs are defined in terms of successor cardinality.
 Any set X with cardinality less than that of the natural numbers ( X  <  N ) is said to be a finite set.
 In set theory, The Hebrew aleph glyph is used as the symbol to denote the aleph numbers, which represent the cardinality of infinite sets.
 The transfinite numbers specify the cardinality of infinite sets.
 Without realizing it, the manager of the Hotel Infinity taught us an important lesson in the cardinality of infinite sets.
 Sameness of cardinality is sometimes referred to as equipotence, equipollence, or equinumerosity.
 This set clearly has cardinality (the natural bijection between the set of binary sequences and P( N) is given by the indicator function).
 Formally, assuming the axiom of choice, cardinality of a set X is the least ordinal α such that there is a bijection between X and α.
 Assuming the axiom of choice holds, the law of trichotomy holds for cardinality, so we have the following definitions.
 Cardinality as defined by my way are more general than cardinality defined by Von Neumann given that the axiom of choice is not in action.
 This is a generalization of the continuum hypothesis since the continuum has the same cardinality as the power set of the integers.
 Cantor's diagonal argument shows that the integers and the continuum do not have the same cardinality.
 The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets.
 The cardinality of the set of natural numbers, Aleph null, has many of the properties of a large cardinal.
 The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers, that is, .
 In mathematics, a countable set is a set with the same cardinality (i.e., number of elements) as some subset of the set of natural numbers.
 It is however possible to discuss the relative cardinality of sets without explicitly assigning names to objects.
 Cantor first established cardinality as an instrument to compare finite sets; e.g.
 For finite sets the cardinality is given by a natural number, being simply the number of elements in the set.
 If a finite set S has cardinality n then the power set of S has cardinality 2 n.
 The concept goes back to Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities.
 Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets.
 The notion of cardinality, as now understood, was formulated by Georg Cantor, the originator of set theory, in 1874 – 1884.
 It is the 2nd beth number, and is the result of cardinal exponentiation when 2 is raised to the power of c, the cardinality of the continuum.
 The second beth number, , is the cardinality of the set of all subsets of the real line.
 The last inequality implies that the cofinality of the cardinality of the continuum must be uncountable.
 The cofinality of A is the least cardinality of a cofinal subset.
 One might naturally wonder whether , the cardinality of the reals, is equal to or if it is strictly larger.
 The cardinality of R is often called the cardinality of the continuum and denoted by c or ( bethone).
 Formally, an uncountable set is defined as one whose cardinality is strictly greater than ( alephnull), the cardinality of the natural numbers.
 Similarly, the question of whether there exists a set whose cardinality is between s and P( s) for some s, leads to the generalized continuum hypothesis.
 The cardinality of the power set is given by the formula 2 N, where N is the number elements in the set.
 This new cardinal number, called the cardinality of the continuum, was termed c by Cantor.
 One can also easily study cardinality without referring to cardinal numbers.
 Alephone is the cardinality of the set of all countably infinite ordinal numbers.
 Any set of cardinality Alephnull can be put into a direct onetoone correspondence (see bijection) with the integers, and thus is a countably infinite set.
 The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers.
 It can be proved that the cardinality of the real numbers is greater than that of the natural numbers just described.
 The latter cardinal number is also often denoted by c; it is the cardinality of the set of real numbers, or the continuum, whence the name.
 Now in this model, all those sets of cardinality kappa which were proper classes in the former theory are now simply sets in this theory.
 Despite the paradoxes and some of the problems one faces in defining and dealing with cardinality, choice, proper classes, etc.
 It is thus said that two sets with the same cardinality are, respectively, equipotent, equipollent, or equinumerous.
 If two sets are given, then they either have the same cardinality, or one has a smaller cardinality than the other.
 For instance, two sets may each have an infinite number of elements, but one may have a greater cardinality.
 In fact, without AC, it's possible to have a family of sets of cardinality three without having a choice function for that family.
 The cardinality of a set A is defined as its equivalence class under equinumerosity.
 I'm not sure why the class of a transitive sets with cardinality omega should not be a set either.
 In the branch of mathematics known as set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets.
 In the branch of mathematics known as set theory, aleph usually refers to a series of numbers used to represent the cardinality (or size) of infinite sets.
 The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite.
 The smallest ordinal having a given cardinal as its cardinality is called the initial ordinal of that cardinal.
 For example, the cardinality of  0 =  is , which is also the cardinality of  or  0 (all are countable ordinals).
 A rooted Kripke frame of has cardinality i , a limit ordinal.
 Any cardinal number can be identified with the smallest ordinal number of cardinality .
 Any set X with cardinality greater than that of the natural numbers ( X   N , for example  R  =  N ) is said to be uncountable.
 Alephtwo is the cardinality of the set of all ordinal numbers of cardinality no greater than alephone.
 For instance, I distinguish a vastly greater number of different infinities than cardinality simply by ordering formulas on a unit infinity.
 Because these sets are not larger than the natural numbers in the sense of cardinality, some may not want to call them uncountable.
 The cardinality of the natural numbers is (alephnull, also alephnaught or alephzero) the next larger cardinality is alephone , then and so on.
 Second, it is false that all sets have a larger cardinality than their proper subsets.
 The concepts of intersection, union, subset, cartesian product and cardinality of multisets are defined by the above formulas.
 Here you can find out something about sets, basic operations on sets, functions, relations, Cartesian product, denumerable sets andcardinality.
 The th infinite initial ordinal is written  . Its cardinality is written .
 The smallest infinite cardinality is that of the natural numbers ().
 The smallest infinite cardinal is the cardinality of a countable set.
 This article incorporates material from cardinality of the continuum on PlanetMath, which is licensed under the GFDL.
 PlanetMath: cardinality of the continuum (Site not responding.
 Each set in this sequence has cardinality strictly greater than the one preceding it, because of Cantor's theorem.
 The nonexistence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis.
 If D is uncountable, but of cardinality strictly smaller than and the poset has the countable chain condition, we can instead use Martin's axiom.
 The lowest transfinite ordinal number is . The first transfinite cardinal number is alephnull, , the cardinality of the infinite set of the integers.
 Cantor called this first transfinite cardinality א 0, It is now known as the continuum, and is denoted by the letter c.
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