Cofinality
Let A be a partially ordered set. A subset B of A is said to be cofinal if for every a in A there is a b in B such that a ≤ b. The cofinality of A is the smallest cardinality of a cofinal subset. Note that the cofinality always exists, since the cardinal numbers are well ordered. Cofinality is only an interesting concept if there is no maximal element in A; otherwise the cofinality is 1.If A admits a totally ordered cofinal subset B, then we can find a subset of B which is well-ordered and cofinal in B (and hence in A). Moreover, any cofinal subset of B whose cardinality is equal to the cofinality of B is well-ordered and order-isomorphic to its own cardinality.
It is a corollary of König's theorem that any well-ordered set with the cardinality of the continuum has uncountable cofinality. Consequently we have
