Forcing with Bushy Trees

TL;DR

This paper explores the combinatorics of bushy trees to establish new results about diagonally noncomputable functions, their computational power, and their relationships with randomness and Turing degrees.

Contribution

It introduces novel results on DNC functions using bushy tree arguments, including their computational limitations and minimal degrees, and surveys related combinatorial techniques.

Findings

Existence of arbitrarily slow-growing DNC functions that do not compute Kurtz random reals.

Construction of DNC functions relative to any oracle with minimal Turing degree.

Survey of bushy tree arguments in computability theory literature.

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.