The purpose of this page is to describe the different types of cardinal numbers. Since all well-ordered cardinals are ordinals, sometimes I use ordinal notation for a cardinal or vice versa. Cardinals A partition of cardinal numbers divides them up into subsets so that each cardinal is a member of one and only one of the subsets. The definitions below help to explain these partitions and the relations between them. Infinite Cardinal A cardinal that is not a finite cardinal is an infinite cardinal.
Every finite cardinal is a Dedekind-finite cardinal.
In ZF it is possible to have a Dedekind-finite cardinal that is not a finite cardinal. Any infinite cardinal that is also an ordinal is an infinite well-ordered cardinal. Any cardinal of the form aleph a where a is an ordinal. Every infinite well-ordered cardinal is an infinite cardinal. In ZF it is possible to have an infinite cardinal which is not a well-ordered cardinal.