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cardinality

See also integer, natural number, rational number, and real number.

The term cardinality refers to the number of cardinal (basic) members in a set. Cardinality can be finite (a non-negative integer) or infinite. For example, the cardinality of the set of people in the United States is approximately 270,000,000; the cardinality of the set of integers is denumerably infinite.

In tables, the number of rows (or tuples) is called the cardinality. In practice, tables always have positive-integer cardinality. The reason for this is simple: tables with no rows, or with a negative number of rows, cannot exist. In theory, however, tables with denumerably infinite cardinality can exist. An example is a multiplication table of non-negative integers in which entries are implied for all possible values:

0 1 2 3 ..
1 1 2 3 ..
2 2 4 6 ..
3 3 6 9 ..
: : : :

The concept of cardinality is of interest to set theoreticians because it has been used to demonstrate that some infinite sets are larger than others. The cardinality of the set of real numbers is greater than the cardinality of the set of integers, even though both sets are infinite. The cardinality of the set of integers is called aleph-null or aleph-nought; the cardinality of the set of real numbers is called aleph-one.

One of the great mysteries of mathematics is contained in the question, "What is the cardinality of the set of points on a geometric line?" Generally it is presumed to be aleph-one; the set of points on a line is thought to correspond one-to-one with the set of real numbers. This is by no means a trivial supposition, and has become known as the Continuum Hypothesis.

This was last updated in September 2005

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