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Published online by Cambridge University Press: 01 May 2018
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 
${\cal S}$.
${\rm{\Pi }}_2^0$ nullsets, this Journal, vol. 71 (2006), no. 3, pp. 1044–1052.Downey,+R.,+Nies,+A.,+Weber,+R.,+and+Yu,+L.,+Lowness+and+${\rm{\Pi+}}_2^0$+nullsets,+this+Journal,+vol.+71+(2006),+no.+3,+pp.+1044–1052.>Google Scholar