Alpha recursion theory: Difference between revisions

Revision as of 02:55, 20 September 2022

Revision as of 14:24, 18 November 2022

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Shore's density theorem: Let ''A'', ''C'' be α-regular recursively enumerable sets such that <math>\scriptstyle A <_\alpha C</math> then there exists a regular α-recursively enumerable set ''B'' such that <math>\scriptstyle A <_\alpha B <_\alpha C</math>.

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]].<!--Barwise. Or 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>

==Relationship to analysis==