Cardinal Number

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A Cardinal Number is a count number used to measure the size of a dataset.



References

2017a

  • (Wikipedia, 2017) ⇒ Retrieved on 2017-06-21 from http://en.wikipedia.org/wiki/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 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 a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets.

      There is a transfinite sequence of cardinal numbers:

[math]0, 1, 2, 3, \ldots, n, \ldots ; \aleph_0, \aleph_1, \aleph_2, \ldots, \aleph_{\alpha}, \ldots.\ [/math]

This sequence starts with the natural numbers including zero (finite cardinals), which are followed by the aleph numbers (infinite cardinals of well-ordered sets). The aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number. If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs.

Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics including model theory, combinatorics, abstract algebra, and mathematical analysis. In category theory, the cardinal numbers form a skeleton of the category of sets.

2017b

  • (Weisstein, 2017) ⇒ Retrieved on 2017-05-21 from: Weisstein, Eric W. "Cardinal Number." From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/CardinalNumber.html
    • In common usage, a cardinal number is a number used in counting (a counting number), such as [math]1, 2, 3, \cdots [/math]

      In formal set theory, a cardinal number (also called “the cardinality") is a type of number defined in such a way that any method of counting sets using it gives the same result. (This is not true for the ordinal numbers.) In fact, the cardinal numbers are obtained by collecting all ordinal numbers which are obtainable by counting a given set. A set has [math]\aleph_0[/math] (aleph-0) members if it can be put into a one-to-one correspondence with the finite ordinal numbers. The cardinality of a set is also frequently referred to as the "power" of a set (Moore 1982, Dauben 1990, Suppes 1972).

      In Georg Cantor's original notation, the symbol for a set [math]A [/math] annotated with a single [math]\overline{A}[/math] indicated [math]A[/math] stripped of any structure besides order, hence it represented the order type of the set. A double overbar [math]\overline{\overline{A}}[/math] then indicated stripping the order from the set and thus indicated the cardinal number of the set. However, in modern notation, the symbol |A| is used to denote the cardinal number of set.

      Cantor, the father of modern set theory, noticed that while the ordinal numbers [math]\omega +1, \omega +2, ...[/math] were bigger than [math]\omega[/math] in the sense of order, they were not bigger in the sense of equipollence. This led him to study what would come to be called cardinal numbers. He called the ordinals [math]\omega, \omega + 1, ...[/math] that are equipollent to the integers "the second number class" (as opposed to the finite ordinals, which he called the "first number class"). Cantor showed

      • 1. The second number class is bigger than the first.
      • 2. There is no class bigger than the first number class and smaller than the second.
      • 3. The class of real numbers is bigger than the first number class.

2017c

  • (Encyclpedia of Mathematice, 2017 ) ⇒ Retrived on 2017-05-21 from "Cardinal number. Encyclopedia of Mathematics". URL: http://www.encyclopediaofmath.org/index.php/Cardinal_number
    • Quote: transfinite number, power in the sense of Cantor, cardinality of a set A

      That property of A that is intrinsic to any set B with the same cardinality as A. In this connection, two sets A and B are said to have the same cardinality (or to be equivalent) if and only if there is a bijective function [math]f:\;A \rightarrow B[/math] with domain A and range B …

2016

  • (Wikinary, 2016) ⇒ http://en.wiktionary.org/wiki/cardinal_number
    • Noun
      • 1. A number used to denote quantity; a counting number.
        • The smallest cardinal numbers are 0, 1, 2, and 3.
        • The cardinal number "three" can be represented as "3" or "three".
      • 2. (mathematics) A generalized kind of number used to denote the size of a set, including infinite sets.
      • 3. (grammar) A word that expresses a countable quantity; a cardinal numeral. "Three" is a cardinal number, while "third" is an ordinal number.

2009