What is effective transfinite recursion in reverse mathematics?

Abstract

In the context of reverse mathematics, effective transfinite recursion refers to a principle that allows us to construct sequences of sets by recursion along arbitrary well orders, provided that each set is $Δ^0_1$-definable relative to the previous stages of the recursion. It is known that this principle is provable in $\mathbf{ACA}_0$. In the present note, we argue that a common formulation of effective transfinite recursion is too restrictive. We then propose a more liberal formulation, which appears very natural and is still provable in $\mathbf{ACA}_0$.