RANDOMNESS IN THE HIGHER SETTING

Abstract We study the strengths of various notions of higher randomness: (i) strong ${\rm{\Pi }}_1^1$ randomness is separated from ${\rm{\Pi }}_1^1$ randomness; (ii) the hyperdegrees of ${\rm{\Pi }}_1^1$ random reals are closed downwards (except for the trivial degree); (iii) the reals z in $NC{R_{{\rm{\Pi }}_1^1}}$ are precisely those satisfying $z \in {L_{\omega _1^z}}$ and (iv) lowness for ${\rm{\Delta }}_1^1$ randomness is strictly weaker than that for ${\rm{\Pi }}_1^1$ randomness.