Alpha recursion theory: Difference between revisions

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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, 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 <math>\alpha</math>-recursive if both A and <math>\alpha \setminus A</math> (its relative complement in <math>\alpha</math>) are <math>\alpha</math> -recursively -enumerable.

Members of <math>L_\alpha</math> are called <math>\alpha</math> finite and play a similar role to the finite numbers in classical recursion theory.