Low (computability)

By the Low basis theorem of Jockusch and Soare, any nonempty $\Pi^0_1$ class in $2^\omega$ contains a set of low degree. This implies that, although low sets are computationally weak, they can still accomplish such feats as computing a completion of Peano Arithmetic. In practice, this allows a restriction on the computational power of objects needed for recursion theoretic constructions: for example, those used in the analyzing the proof-theoretic strength of Ramsey's theorem.