Weakly 2-randoms and 1-generics in scott sets

Let S be a Scott set, or even an ω-model of WWKL. Then for each A ∈ S, either there is X ∈ S that is weakly 2-random relative to A, or there is X ∈ S that is 1-generic relative to A. It follows that if A₁,⋯, A_n ∈ S are noncomputable, there is X ∈ S such that each A_i is Turing incomparable with X, answering a question of Kučera and Slaman.More generally, any ∀∃ sentence in the language of partial orders that holds inD also holds in D^S, where D^S is the partial order of Turing degrees of elements of S.