The many faces of omega-logic

We consider several formalizations in the language of second-order arithmetic of "The formula $\phi$ is a theorem of $\omega$-logic", including some which have been studied in the literature and a new variant defined via a least fixed point. We analyze the provability of relations between these different formalizations in standard theories of reverse mathematics. With this, we study the strength of various reflection principles arising from these notions of provability, surveying known results and establishing some new equivalences, including a characterization of $\Pi^1_1$-${\sf CA}_0$ in terms of our fixed-point formalization of $\omega$-logic.