I'm sure this is a fairly basic question, but I can't seem to find a solid answer:


My primary question is: Is there a reasonably nice subsystem of second-order arithmetic corresponding essentially to "primitive recursive comprehension?" I'm interested only in $\omega$-models - that is, I don't care about how much induction the system allows. In fact, I'd prefer it to have full induction, so that I know the weakness of the system lies squarely in its comprehension axioms. I do mean "comprehension" here: I'd like this system to be a two-sorted system, like $RCA$ itself, so e.g. $PRA$ is not what I'm looking for - although maybe it can be tweaked into a satisfactory system in an easy way?


My secondary question is: assuming a positive answer to the first question, what are some natural statements which are equivalent (over this base system) to $RCA$ (or $RCA\_0$)?


(I'm almost certain this is written up nicely somewhere easily accessible, and my google-fu is simply failing me; if this is the case, and this question is therefore inappropriate for mathoverflow, please feel free to close it.)



Reverse mathematics below RCA
It looks like there is now a paper explicitly treating such a system! [Bazhenov/Fiori-Carones/Liu/Melnikov, *Primitive Recursive Reverse Mathematics*](https://arxiv.org/pdf/2210.13080.pdf) just appeared on the arxiv. Here are a couple key points:


* Models of their base theory, $\mathsf{PRA}^2$, are precisely pairs $(M,X)$ where $M\models\mathsf{PRA}$ and $X$ is a family of functions on $M$ closed under composition and primitive recursion and containing the constants.
* Over $\mathsf{PRA^2}$, there are two natural versions of recursive comprehension: $\mathsf{RCA\_0}$ itself and the somewhat-weaker $2^\mathbb{N}$-$\mathsf{RCA\_0}$. The difference is whether the objects considered belong to Baire or Cantor space.


