**Kleene's**

The basic idea of Kleene's system of ordinal notations is to build up ordinals in an effective manner. For members of, the ordinal for which is a notation is . The standard definition proceeds via transfinite induction and the ordering (a partial ordering of Kleene's ) is defined simultaneously.

- The natural number 0 belongs to Kleene's and .
- If belongs to Kleene's and, then belongs to Kleene's and and .
- Suppose is the -th partial recursive function. If is total, with range contained in, and for every natural number, we have, then belongs to Kleene's, for each and, i.e. is a notation for the limit of the ordinals where for every natural number .
- and imply (this guarantees that is transitive.)

This definition has the advantages that one can recursively enumerate the predecessors of a given ordinal (though not in the ordering) and that the notations are downward closed, i.e., if there is a notation for and then there is a notation for .

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