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RANDOMNESS IN THE HIGHER SETTING

Published online by Cambridge University Press:  22 December 2015

C. T. CHONG  and
LIANG YU
C. T. CHONG
Affiliation:
DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE NATIONAL UNIVERSITY OF SINGAPORE SINGAPORE 119076, SINGAPOREE-mail: [email protected]
LIANG YU
Affiliation:
INSTITUTE OF MATHEMATICAL SCIENCE NANJING UNIVERSITY, JIANGSU PROVINCE 210093 P. R. OF CHINAE-mail: [email protected]
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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.

Keywords

03D3003D3268Q30Hyperarithmetic theoryhyperdegreesΠ11 randomnessΠ11 Martin-Löf randomnesslowness for higher randomness
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 4 , December 2015 , pp. 1131 - 1148
Copyright
Copyright © The Association for Symbolic Logic 2015 

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