Alpha recursion theory: Difference between revisions

Revision as of 02:56, 3 November 2021

Revision as of 06:17, 5 December 2021

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We say ''A'' is regular if <math>\forall \beta \in \alpha: A \cap \beta \in L_\alpha</math> or in other words if every initial portion of ''A'' is α-finite.

==Results in <math>\alpha</math>α recursion==

Shore's splitting theorem: Let A be <math>\alpha</math> recursively enumerable and regular. There exist <math>\alpha</math> recursively enumerable <math>B_0,B_1</math> such that <math>A=B_0 \cup B_1 \wedge B_0 \cap B_1 = \varnothing \wedge A \not\le_\alpha B_i (i<2).</math>

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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)-->

==Relation to analysis==

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.<!--https://arxiv.org/pdf/1808.03814.pdf#page=4, "P_α = P(N) ∩ L_α"-->