The cofinality of an infinite cardinal, seen as an ordinal with the natural well-order, is itself. |
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.