Get Recursive Ordinal essential facts below. View Videos or join the Recursive Ordinal discussion. Add Recursive Ordinal to your PopFlock.com topic list for future reference or share this resource on social media.
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.