Let $\emptyset^{(n)}$ denote the $n$th Turing jump of the empty set. Post's theorem states:
A set $B$ is $\Sigma^0_{n+1}$ if and only if $B$ is recursively enumerable by an oracle Turing machine with an oracle for $\emptyset^{(n)}$, that is, if and only if $B$ is $\Sigma^{0,\emptyset^{(n)}}_{1}$.
This is a very powerful correspondence and I would like to use it or something similar in my work on hyperarithmetic sets. Is there a generalization of Post's theorem that also describes infinitely iterated Turing jumps, i.e. where we replace $n$ by an ordinal $\alpha < \omega_1^{CK}$? Why or why not is it possible to relate $\emptyset^{(\omega^2)}$ with $\Sigma^{0}_{\omega^2+1}$, for example?