Turing Incomparability in Scott Sets

Weakly 2-randoms and 1-generics in Scott sets

Let $S$ be a Scott set, or even an $ω$-model of $\mathsf{WWKL}$. Then for each $A\in S$, either there is $X \in S$ that is weakly 2-random relative to $A$, or there is $X\in S$ that is 1-generic relative to $A$. It follows that if $A_1,\dots, A_n \in S$ are non-computable, there is $X \in S$ such that each $A_i$ is Turing incomparable with $X$, answering a question of Kučera and Slaman. More generally, any $\forall\exists$ sentence in the language of partial orders that holds in $\mathcal D$ also holds in $\mathcal D_S$, where $\mathcal D_S$ is the partial order of Turing degrees of elements o…