What is effective transfinite recursion in reverse mathematics?

TL;DR

This paper examines the principle of effective transfinite recursion in reverse mathematics, proposing a more flexible formulation that remains provable in CA_0, and clarifies its foundational role.

Contribution

It introduces a more general formulation of effective transfinite recursion that broadens its applicability while maintaining provability in CA_0.

Findings

The common formulation of effective transfinite recursion is too restrictive.

A new, more liberal formulation is proposed.

The new formulation is still provable in CA_0.

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 $\Delta^0_1$-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 $\mathbf{ACA}_0$.