Jump Inversion of Arithmetic

This doesn't exactly answer your question but...

If $A$ is any upper bound for the arithmetic degrees then $0^{(\omega)}$ is recursive in $A^{\prime\prime}$. Enderton and Putnam proved that there upper bounds with $A^{\prime\prime}=0^{(\omega)} $

Dave

As far as actual jump inversion goes, any degree $X \geq 0^{(n)}$ is the $n$th jump of some other degree. An easy way to see this is to apply Friedberg jump inversion relative to $0^{(n-1)}$, then relative to $0^{(n-2)}$, and so on down to $0$. The theorem is also true through transfinite iterates of the jump: if $X \geq 0^{(\alpha)}$, then $X$ is the $\alpha$th jump of some degree. The general version of this theorem for any $\alpha$-REA operator is due to Jockusch and Shore (1984).