Foundations of Mathematics
What are cardinal and ordinal numbers and finite sets in set theory?
Cardinal and ordinal numbers are used in reference to numbers: Ordinal numbers are used to describe the position of objects or entities arranged in a certain sequence, such as first, second, third, and so on; cardinal numbers are natural numbers, or 0, 1, 2, 3, and so on. (For more information about cardinal and ordinal numbers, see “Math Basics.”)
But cardinal numbers used in set theory describe the number of members in a set. Both ordinal and cardinal numbers are further used to describe infinite sets, and are prefaced with “first ordinal” or “first cardinal” infinities. The first ordinal infinity applies to the smallest number greater than any finite ordered set of natural numbers. The first cardinal infinity applies to the number of all the natural numbers. (For more about infinite sets, see below.)
A finite set is one that is not infinite; it can be numbered from 1 to n, for some positive integer n. This number n is also called the set’s cardinal number; thus, for a certain set A, the cardinality is denoted by card(A). There are a number of rules to cardinal numbers and finite sets. For example, if two sets bisect, then they are said to have the same cardinality (or power). The empty set is considered to be a finite set, with its set’s cardinal number being 0.