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In [[recursion theory]], '''α recursion theory''' is a generalisation of [[recursion theory]] to subsets of [[admissible ordinal]]s <math>\alpha</math>. An admissible set is closed under <math>\Sigma_1(L_\alpha)</math> functions<ref>K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf ''An introduction to the fine structure of the constructible hierarchy''] (p.38)</ref>, where <math>L_\xi</math> denotes a rank of Godel's [[constructible hierarchy]]. If <math>L_{\alpha}</math> is a model of [[Kripke–Platek set theory]] then <math>\alpha</math> is an admissible ordinal. In what follows <math>\alpha</math> is considered to be fixed.
The objects of study in <math>\alpha</math> recursion are subsets of <math>\alpha</math>. A is said to be '''<math>\alpha</math> recursively enumerable''' if it is <math> \Sigma_1</math> definable over <math>L_\alpha</math>. A is recursive if both A and <math>\alpha \setminus A</math> (its complement in <math>\alpha</math>) are <math>\alpha</math> recursively enumerable.
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