This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member. The cofinality of a partially ordered set A can alternatively be defined as the least ordinal x such that there is a function from x to A with cofinal image. This second definition makes sense without the axiom of choice. If the axiom of choice is assumed, as will be the case in the rest of this article, then the two definitions are equivalent.
This definition of cofinality relies on the axiom of choice as it uses the fact that every non-empty set of cardinal numbers has a least member The cofinality of a partially ordered set A can alternatively be defined as the least ordinal x such that there is a function from x to A with cofinal image This second definition makes sense without the axiom of choice If the axiom of choice is assumed as will be the case in the rest of this article then the two definitions are equivalent