RANDOMNESS VIA INFINITE COMPUTATION AND EFFECTIVE DESCRIPTIVE SET THEORY

D ec 2 01 6 RANDOMNESS VIA INFINITE COMPUTATION AND EFFECTIVE DESCRIPTIVE SET THEORY

We study randomness beyond Π11-randomness and its Martin-Löf type variant, introduced in [HN07] and further studied in [BGM]. The class given by the infinite time Turing machines (ITTMs), introduced by Hamkins and Kidder, is strictly between Π11 and Σ 1 2. We prove that the natural randomness notions associated to this class have several desirable properties resembling those of the classical ra...

Randomness via effective descriptive set theory

1.1 Informal introduction to Π1 relations Let 2N denote Cantor space. • A relation B ⊆ N × (2N)r is Π1 if it is obtained from an arithmetical relation by a universal quantification over sets. • If k = 1, r = 0 we have a Π1 set ⊆ N. • If k = 0, r = 1 we have a Π1 class ⊆ 2N. • The relation B is∆1 if both B and its complement are Π1. There is an equivalent representation of Π1 relations where the...

Randomness in effective descriptive set theory

An analog of ML-randomness in the effective descriptive set theory setting is studied, where the r.e. objects are replaced by their Π1 counterparts. We prove the analogs of the Kraft-Chaitin Theorem and Schnorr’s Theorem. In the new setting, while K-trivial sets exist that are not hyper-arithmetical, each low for random set is. Finally we study a very strong yet effective randomness notion: Z i...

Intuitionism and Effective Descriptive Set Theory

Notes on Effective Descriptive Set Theory