Cardinality is a concept in set theory to describe the size of any set. Ordinal numbers represent order.

In mathematics, cardinality and ordinal number are concepts used to describe the number of elements in a set or the order relationship between elements. Cardinality refers to the number of elements in a set, that is, the size of the set. The symbol |S| is used to indicate the cardinality of the set S. For example, the cardinality of the set {1, 2,3} is 3. Ordinal number is used to describe the order relationship of elements in a set, indicating the position of elements in an ordered set. In ordinal number, the ordinal number of the first element is 1, the ordinal number of the second element is 2, and so on.

It should be noted that cardinality and ordinal number are related to the number of elements in a set, but they describe different concepts. Cardinality focuses on the number of elements, and ordinal focuses on the position of elements. In a set, the position of elements can be changed at will, but the number of elements is constant, so cardinality and ordinal number are two different concepts.

Mathematical definition of ordinal number

Ordinal number was originally defined as ordinal number type of well-ordered set, and ordinal number type of well-ordered set A, as the result abstracted from the attributes of elements of A, is defined as {B|BA} because all well-ordered sets are isomorphic to the * * same characteristics of order A.

This definition is very simple and clear in form, but it cannot be proved that it constitutes a ZFC axiomatic system. In fact, {B|BA} is a real class. Therefore, the original definition is unsuccessful and must be revised and adopted in another way. Let α be a well-ordered set, ξ ∈ α, and say that S (ξ) = {β ∈ α | β < ξ} is the initial line segment generated by ξ in a well-ordered set α.

The above contents refer to Baidu Encyclopedia-ordinal number