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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>.<ref>T. Arai, [https://www.sciencedirect.com/science/article/pii/S0168007203000204 Proof theory for theories of ordinals - I: recursively Mahlo ordinals] (p.2). Accessed 2021-07-29.</ref>
==References==
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