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Recursion Theory by Liang Yu, Chi Tat Chong

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14 The theory of higher randomness

Higher randomness theory studies algorithmic complexity problems of randomness using techniques and ideas from hyperarithmetic theory and set theory. Since second order arithmetic provides a broader, though sometimes coarser, view of the reals, the picture one gets of randomness is quite different from that for first order theory.

14.1 Higher Kurtz randomness

14.1.1 image and image-Kurtz randomness

Definition 14.1.1. Let x ∈ 2ω. Given a class Γ of sets of reals, x is Γ-Kurtz random if it is not a member of any closed nullset ...

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