A measure of the number of elements in a group or a set. For example, the number of books on a shelf can be described by a single cardinal number. Similarly, the set can be assigned the cardinal number 3 because it has only three elements. Since cardinal numbers count the number of elements in a set, they are always positive whole integers. If the elements from two sets have a one-to-one relationship, namely each element can be paired together such that no elements are left over, then they can be represented by the same cardinal number.
Some sets have an infinite number of elements. However, not all infinite sets have a one-to-one relationship. Consider the following sets:
Set X Set Y Although both of these sets have an infinite number of elements, they can not be represented by the same cardinal number because set X contains all the elements of set Y, but it also contains additional elements. To solve this problem a nineteenth century mathematician named George Cantor (1845-1918) created a new numbering system to deal with infinite sets. He called these new numbers transfinite cardinal numbers and used the symbol u0 (aleph null) to represent the smallest one. He also developed an arithmetic system for manipulating these numbers.