Set-theoretic reflection is equivalent to induction over well-founded classes

TL;DR

This paper establishes an equivalence between set-theoretic reflection principles and induction over well-founded classes definable by certain formulas, within a weak foundational framework.

Contribution

It proves that reflection principles are equivalent to induction over $ riangle( eals)$-definable well-founded classes in primitive recursive set theory with dependent choice.

Findings

Reflection principle is equivalent to induction over well-founded classes.

Equivalence holds in primitive recursive set theory extended by dependent choice.

Results connect reflection principles with induction in a weaker set-theoretic context.

Abstract

We show that induction over $\Delta(\mathbb R)$-definable well-founded classes is equivalent to the reflection principle which asserts that any true formula of first order set theory with real parameters holds in some transitive set. The equivalence is proved in primitive recursive set theory (which is weaker than Kripke-Platek set theory) extended by the axiom of dependent choice.