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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]].<!--Barwise. Or T. Arai, [https://www.sciencedirect.com/science/article/pii/S0168007203000204 Proof theory for theories of ordinals - I: recursively Mahlo ordinals] (p.2)-->
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Some results in <math>\alpha</math>-recursion can be translated into similar results about [[second-order arithmetic]]. This is because of the relationship <math>L</math> has with the ramified analytic hierarchy, an analog of <math>L</math> for the language of second-order arithmetic, that consists of sets of integers.<ref>P. D. Welch, [https://arxiv.org/pdf/1808.03814.pdf#page=4 The Ramified Analytical Hierarchy using Extended Logics] (2018, p.4). Accessed 8 August 2021.</ref><!--"P_α = P(N) ∩ L_α"-->
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