Cardinal number

In set theory, cardinality is a property of sets that generalises the notion of the size of a finite set. Because of this, it is often thought of as the “size” of a (possibly infinite) set. While this is useful for thinking about the concept, it can lead to misconceptions. Some of the intuitive notions associated with size do not carry over to cardinality, and some do in certain set theories but not in others.

Motivation

In comparing two finite sets, say {1, 2, 3} and {a, b, c, d, e}, the elements may be counted to give a unique natural number for each set, here 3 and 5 respectively, called the size of the set. 5 is a greater number than 3, and so the second set is said to be greater than the first. Various properties of size are intuitively familiar, such as the fact that adding elements to a set yields a set of greater size, while removing elements yields a set of smaller size. But when considering two infinite sets, say the set of natural numbers {1, 2, 3, 4,... } and the set of natural numbers greater than 19, {20, 21, 22,... }, the second is obtained by removing elements from the first, but there is no obvious way of assigning a “number” to each of the sets which would indicate that the second is “smaller” than the first.

Therefore, instead of attempting to generalise the process of counting, mathematicians generalise another relation between finite sets that is equivalent to their being the same size. Two finite sets X and Y are the same size precisely if the elements of X can be mapped to the elements of Y in such a way that every element in Y is mapped to by exactly one element in X; in this case X and Y are said to be equinumerous or equipotent. Finiteness is not necessary to ask whether two sets are equinumerous so we can immediately generalise it to arbitrary pairs of sets.

Definition

The cardinality of a set X then, will be an object associated with X, which is denoted |X|, that should satisfy the property:

|X| = |Y| if and only if X and Y are equinumerous.

There may be many such objects, any of which could be used as a definition of cardinality. The objects chosen to be used as the cardinality of sets are called cardinals or cardinal numbers. Which possible definitions are available for use will depend on the set theory one is working with. The collection of sets equinumerous with X is the same as the collection equinumerous with Y precisely when X and Y are themselves equinumerous. If therefore there is a set whose elements are precisely the sets equinumerous with X, this is usually taken as the definition of |X|. However the existence of such sets is not consistent with Zermelo set theory and its extentions, which are the most commonly used set theories. However, there are workarounds, such as using the set of all sets of minimal rank equinumerous with X. When the axiom of choice is available, X can always be well ordered, and |X| can be defined as the least ordinal that is the order type of some well ordering of X; this is the definition of cardinal numbers most familiar to mathematicians.