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WEAKLY 2-RANDOMS AND 1-GENERICS IN SCOTT SETS
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- Journal:
- The Journal of Symbolic Logic / Volume 83 / Issue 1 / March 2018
- Published online by Cambridge University Press:
- 01 May 2018, pp. 392-394
- Print publication:
- March 2018
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- Article
- Export citation
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Let
${\cal S}$ be a Scott set, or even an ω-model of WWKL. Then for each A ε S, either there is X ε S that is weakly 2-random relative to A, or there is X ε S that is 1-generic relative to A. It follows that if A1,…,An ε S are noncomputable, there is X ε S such that each Ai is Turing incomparable with X, answering a question of Kučera and Slaman. More generally, any ∀∃ sentence in the language of partial orders that holds in 
${\cal D}$ also holds in 
${{\cal D}^{\cal S}}$, where 
${{\cal D}^{\cal S}}$ is the partial order of Turing degrees of elements of 
${\cal S}$.
