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1 results

RANDOMNESS IN THE HIGHER SETTING
C. T. CHONG, LIANG YU
Journal:

The Journal of Symbolic Logic / Volume 80 / Issue 4 / December 2015

Published online by Cambridge University Press:

22 December 2015, pp. 1131-1148

Print publication:

December 2015
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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.