In recursion theory, **α recursion theory** is a generalisation of recursion theory to subsets of admissible ordinals . An admissible set is closed under functions. If is a model of Kripke–Platek set theory then is an admissible ordinal. In what follows is considered to be fixed.

The objects of study in recursion are subsets of . A is said to be ** recursively enumerable** if it is definable over . A is recursive if both A and (its complement in ) are recursively enumerable.

Members of are called finite and play a similar role to the finite numbers in classical recursion theory.

We say R is a reduction procedure if it is recursively enumerable and every member of R is of the form where *H*, *J*, *K* are all α-finite.

*A* is said to be α-recursive in *B* if there exist reduction procedures such that:

If *A* is recursive in *B* this is written . By this definition *A* is recursive in (the empty set) if and only if *A* is recursive. However A being recursive in B is not equivalent to A being .

We say *A* is regular if or in other words if every initial portion of *A* is α-finite.

## Results in recursion

Shore's splitting theorem: Let A be recursively enumerable and regular. There exist recursively enumerable such that

Shore's density theorem: Let *A*, *C* be α-regular recursively enumerable sets such that then there exists a regular α-recursively enumerable set *B* such that .

## References

- Gerald Sacks,
*Higher recursion theory*, Springer Verlag, 1990 https://projecteuclid.org/euclid.pl/1235422631 - Robert Soare,
*Recursively Enumerable Sets and Degrees*, Springer Verlag, 1987 https://projecteuclid.org/euclid.bams/1183541465