There is an extremely rich and well-understood analogy between "recursively enumerable" and "$\Pi^1_1$" - indeed, this is the starting point of metarecursion theory, and $\alpha$-recursion theory in general (see Sacks' wonderful book http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.pl/1235422632&page=record).

I'm interested in whether there is a straightforward reverse mathematical analogy to be made here, as well. Specifically, we usually think of $RCA_0$ and $ACA_0$ as "recursive comprehension" and "recursively enumerable comprehension," respectively; I'm wondering if there is a theory $T$ such that we can - in a similar way - think of $T$ as "metarecursive comprehension" and $\Pi^1_1-CA_0$ as "metarecursively enumerable comprehension."

A natural candidate for that theory might be $\Delta^1_1-CA_0$, but that's not quite right: in the metarecursive context, $\Delta^1_1$ corresponds to finite (since intuitively everything is relative to Kleene's $\mathcal{O}$). I've been thinking that the right $T$ here is $ATR_0$, but I don't really have any serious reason to back that up.

My instinct is that this is not really the right question - metarecursion theory doesn't live on $\omega$, it lives on $\omega_1^{CK}$, so the right thing to do is formulate a version of reverse math on $\omega_1^{CK}$ (in fact, Richard Shore http://www.math.cornell.edu/~shore/papers/pdf/RMComp7.pdf has already done this for cardinals, and I don't see anything preventing his work extending to arbitrary admissible ordinals - although his theorems might not, of course, since they may require more admissiblity than $\Sigma_1$). But I'm still holding out hope that there's something neat that can happen already on $\omega$.

(I've tagged "descriptive-set-theory" because of the deep connections between higher recursion theory and descriptive set theory, but if anyone feels that that's too much of a reach here, feel free to untag.)