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In recursion theory, α recursion theory is a generalisation of recursion theory to subsets of admissible ordinals ${\displaystyle \alpha }$. An admissible ordinal is closed under ${\displaystyle {\Sigma }_1(L_{\alpha })}$ functions. Admissible ordinals are models of Kripke–Platek set theory. In what follows ${\displaystyle \alpha }$ is considered to be fixed.

The objects of study in ${\displaystyle \alpha }$ recursion are subsets of ${\displaystyle \alpha }$. A is said to be ${\displaystyle \alpha }$ recursively enumerable if it is ${\displaystyle {\Sigma }_1}$ definable over ${\displaystyle L_{\alpha }}$. A is recursive if both A and ${\displaystyle \alpha /A}$ (its complement in ${\displaystyle \alpha }$) are ${\displaystyle \alpha }$ recursively enumerable.

Members of ${\displaystyle L_{\alpha }}$ are called ${\displaystyle \alpha }$ 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 ${\displaystyle ⟨H,J,K⟩}$ where H, J, K are all α-finite.

A is said to be α-recusive in B if there exist ${\displaystyle R_0,R_1}$ reduction procedures such that:

${\displaystyle K\subseteq A\leftrightarrow \mathrm{\exists }H:\mathrm{\exists }J:[⟨H,J,K⟩\in R_0\wedge H\subseteq B\wedge J\subseteq \alpha /B],}$

${\displaystyle K\subseteq \alpha /A\leftrightarrow \mathrm{\exists }H:\mathrm{\exists }J:[⟨H,J,K⟩\in R_1\wedge H\subseteq B\wedge J\subseteq \alpha /B].}$

If A is recursive in B this is written ${\displaystyle A{\le }_{\alpha }B}$. By this definition A is recursive in ${\displaystyle \varnothing }$ (the empty set) if and only if A is recursive. However A being recursive in B is not equivalent to A being ${\displaystyle {\Sigma }_1(L_{\alpha }[B])}$.

We say A is regular if ${\displaystyle \mathrm{\forall }\beta \in \alpha :A\cap \beta \in L_{\alpha }}$ or in other words if every initial portion of A is α-finite.

Shore's splitting theorem: Let A be ${\displaystyle \alpha }$ recursively enumerable and regular. There exist ${\displaystyle \alpha }$ recursively enumerable ${\displaystyle B_0,B_1}$ such that ${\displaystyle A=B_0\cup B_1\wedge B_0\cap B_1=\varnothing \wedge A{\le }_{\alpha }{B}_i(i<2).}$

Shore's density theorem: Let A, C be α-regular recursively enumerable sets such that ${\displaystyle A{<}_{\alpha }C}$ then there exists a regular α-recursively enumerable set B such that ${\displaystyle A{<}_{\alpha }B{<}_{\alpha }C}$.

• Gerald Sacks, Higher recursion theory, Springer Verlag, 1990
• Robert Soare, Recursively Enumerable Sets and Degrees, Springer Verlag, 1987