Alpha recursion theory: Difference between revisions

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Barwise has proved that the sets <math>\Sigma_1</math>-definable on <math>L_{\alpha^+}</math> are exactly the sets <math>\Pi_1^1</math>-definable on <math>L_\alpha</math>, where <math>\alpha^+</math> denotes the next admissible ordinal above <math>\alpha</math>, and <math>\Sigma</math> is from the [[Levy hierarchy]].<ref>T. Arai, [https://www.sciencedirect.com/science/article/pii/S0168007203000204 Proof theory for theories of ordinals - I: recursively Mahlo ordinals] (1998). p.2</ref>

There is a generalization of [[Computability in the limit|limit computability]] to partial <math>\alpha\to\alpha</math> functions.<math>\alpha</math>.<ref>S. G. Simpson, "Degree Theory on Admissible Ordinals", pp.170--171. Appearing in J. Fenstad, P. Hinman, ''Generalized Recursion Theory: Proceedings of the 1972 Oslo Symposium'' (1974), ISBN 0 7204 22000.</ref>

==Relationship to analysis==