  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.
