Forcing with Bushy Trees

Abstract

We present several results that rely on arguments involving the combinatorics of "bushy trees". These include the fact that there are arbitrarily slow-growing diagonally noncomputable (DNC) functions that compute no Kurtz random real, as well as an extension of a result of Kumabe in which we establish that there are DNC functions relative to arbitrary oracles that are of minimal Turing degree. Along the way, we survey some of the existing instances of bushy tree arguments in the literature.