Alpha recursion theory: Difference between revisions

Revision as of 19:46, 8 August 2021

Revision as of 19:46, 8 August 2021

Line 25:

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)-->

==Relation to analysis==

Some results in <math>\alpha</math>-recursion can be reshapedtranslated 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.<!--https://arxiv.org/pdf/1808.03814.pdf#page=4, "P_α = P(N) ∩ L_α"-->

==References==