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When two sets, and , have the same cardinality, it is usually written as | | = | |; however, if referring to the cardinal number of an individual set , it is simply denoted | |, with a vertical bar on each side; [3] this is the same notation as absolute value, and the meaning depends on context.
The new cardinal number of the set of real numbers is called the cardinality of the continuum and Cantor used the symbol for it. Cantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallest transfinite cardinal number ( ℵ 0 {\displaystyle \aleph _{0}} , aleph-null), and that for every ...
They were introduced by the mathematician Georg Cantor [1] and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ). [2] [a] The cardinality of the natural numbers is ℵ 0 (read aleph-nought, aleph-zero, or aleph-null), the next larger cardinality of a well-ordered set is aleph-one ℵ 1, then ℵ 2 and so on.
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for ...
The category < of sets of cardinality less than and all functions between them is closed under colimits of cardinality less than . κ {\displaystyle \kappa } is a regular ordinal (see below). Crudely speaking, this means that a regular cardinal is one that cannot be broken down into a small number of smaller parts.
The cardinality of the continuum is the size of the set of real numbers. The continuum hypothesis is sometimes stated by saying that no cardinality lies between that of the continuum and that of the natural numbers , ℵ 0 {\displaystyle \aleph _{0}} , or alternatively, that c = ℵ 1 {\displaystyle {\mathfrak {c}}=\aleph _{1}} .
Cardinal functions are widely used in topology as a tool for describing various topological properties. [2] [3] Below are some examples.(Note: some authors, arguing that "there are no finite cardinal numbers in general topology", [4] prefer to define the cardinal functions listed below so that they never taken on finite cardinal numbers as values; this requires modifying some of the ...
Beth numbers are defined by transfinite recursion: =, + =, = {: <}, where is an ordinal and is a limit ordinal. [1]The cardinal = is the cardinality of any countably infinite set such as the set of natural numbers, so that = | |.