It is easy to check that is recursive. The successor of a recursive ordinal is recursive, and the set of all recursive ordinals is closed downwards.
The supremum of all recursive ordinals is called the Church–Kleene ordinal and denoted by . The Church–Kleene ordinal is a limit ordinal. An ordinal is recursive if and only if it is smaller than . Since there are only countably many recursive relations, there are also only countably many recursive ordinals. Thus, is countable.