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We use concepts of continuous higher randomness, developed in Bienvenu et al. [‘Continuous higher randomness’, J. Math. Log. 17(1) (2017).], to investigate $\unicode[STIX]{x1D6F1}_{1}^{1}$ -randomness. We discuss lowness for $\unicode[STIX]{x1D6F1}_{1}^{1}$ -randomness,… Click to show full abstract
We use concepts of continuous higher randomness, developed in Bienvenu et al. [‘Continuous higher randomness’, J. Math. Log. 17(1) (2017).], to investigate $\unicode[STIX]{x1D6F1}_{1}^{1}$ -randomness. We discuss lowness for $\unicode[STIX]{x1D6F1}_{1}^{1}$ -randomness, cupping with $\unicode[STIX]{x1D6F1}_{1}^{1}$ -random sequences, and an analogue of the Hirschfeldt–Miller characterization of weak 2-randomness. We also consider analogous questions for Cohen forcing, concentrating on the class of $\unicode[STIX]{x1D6F4}_{1}^{1}$ -generic reals.
Journal Title: Forum of Mathematics, Sigma
Year Published: 2017
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