A $\Pi^0_2$ Singleton of Minimal Arithmetic Degree

Abstract

In the study of the arithmetic degrees (the degree structure induced by relative arithmetic definability, ($\leq_{a}$) the $\omega$-REA sets play a role analogous to the role the r.e. degrees play in the study of the Turing degrees. However, much less is known about the arithmetic degrees and the role of the $\omega$-REA sets in that structure than about the Turing degrees. Indeed, even basic questions such as the existence of a $\omega$-REA set of minimal arithmetic degree are open. This paper makes progress on this question by demonstrating that some promising approaches inspired by the analogy with the r.e sets fail to show that no $\omega$-REA set is arithmetically minimal. Finally, it constructs a $\Pi^0_2$ singleton of minimal arithmetic degree. Not only is this a result of considerable interest in it's own right, constructions of $\Pi^0_2$ singletons often pave the way for constructions of $\omega$-REA sets with similar properties. Along the way, a number of interesting results relating arithmetic reducibility and rates of growth are established.