I seem to recall once hearing a result to the effect that $\emptyset^{(\omega)}$ was the double jump of some other degree, but could not be the triple jump of any degree. However I'm unable to find the exact result. Does anyone know what I might be thinking of (or what is actually known about jump inversion on $\emptyset^{(\omega)}$, if I'm remembering this completely wrong)?



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


