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 S0 is a Scott set (i.e. an ω-model of weak König’s lemma) and A ∈ S0, A ⊆ ω, is incomputable, then there exists B ∈ S0, B ⊆ ω, such that A and B are Turing incomparable. 1. Genericity, Randomness, Logic, and Computability 1.