A measure-theoretic proof of Turing incomparability

We prove that if S is an ω -model of weak weak König’s lemma and A ∈ S , A ⊆ ω , is incomputable, then there exists B ∈ S , B ⊆ ω , such that A and B are Turing incomparable. This extends a recent result of Kučera and Slaman who proved that if S 0 is a Scott set (i.e. an ω -model of weak König’s lemma) and A ∈ S 0 , A ⊆ ω , is incomputable, then there exists B ∈ S 0 , B ⊆ ω , such that A and B are Turing incomparable.