In recursion theory, α recursion theory is a generalisation of recursion theory to subsets of admissible ordinals .  An admissible set is closed under  functions, where  denotes a rank of Godel's constructible hierarchy.  is an admissible ordinal if  is a  model of Kripke–Platek set theory.  In what follows  is considered to be fixed.

The objects of study in  recursion are subsets of . These sets are said to have some properties:
A set  is said to be -recursively-enumerable if it is  definable over , possibly with parameters from  in the definition.
A is -recursive if both A and  (its relative complement in ) are -recursively-enumerable. It's of note that -recursive sets are members of  by definition of .
Members of  are called -finite and play a similar role to the finite numbers in classical recursion theory.
Members of  are called -arithmetic. 

There are also some similar definitions for functions mapping  to :
A function mapping  to  is -recursively-enumerable, or -partial recursive, iff its graph is -definable on .
A function mapping  to  is -recursive iff its graph is -definable on .
Additionally, a function mapping  to  is -arithmetical iff there exists some  such that the function's graph is -definable on .

Additional connections between recursion theory and α recursion theory can be drawn, although explicit definitions may not have yet been written to formalize them:
The functions -definable in  play a role similar to those of the primitive recursive functions.

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 α 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 .

Barwise has proved that the sets -definable on  are exactly the sets -definable on , where  denotes the next admissible ordinal above , and  is from the Levy hierarchy.

There is a generalization of limit computability to partial  functions..

Relationship to analysis
Some results in -recursion can be translated into similar results about second-order arithmetic. This is because of the relationship  has with the ramified analytic hierarchy, an analog of  for the language of second-order arithmetic, that consists of sets of integers.

In fact, when dealing with first-order logic only, the correspondence can be close enough that for some results on , the arithmetical and Levy hierarchies can become interchangeable. For example, a set of natural numbers is definable by a  formula iff it's -definable on , where  is a level of the Levy hierarchy. More generally, definability of a subset of ω over HF with a  formula coincides with its arithmetical definability using a  formula.

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
 Keith J. Devlin, An introduction to the fine structure of the constructible hierarchy (p.38), North-Holland Publishing, 1974
 J. Barwise, Admissible Sets and Structures. 1975

Inline references

Computability theory