Is it known if $0^{(\omega)}$ a minimal cover in the arithmetic degrees. In the Turing degrees to show that $0'$ (indeed $0^{(n)}$) isn't a minimal cover one uses the density of r.e. degrees. However, given that it's not known if downward density holds for the REA sets (even in the arithmetic degrees) is it known if $0^{(\omega)}$ a minimal cover in the arithmetic degrees.