A cumulative hierarchy of sets for constructive set theory

Abstract

The von Neumann hierarchy of sets is heavily used as a basic tool in classical set theory, being an underlying ingredient in many proofs and concepts. In constructive set theories like without the powerset axiom however, it loses much of its potency by ceasing to be a hierarchy of sets as its single stages become only classes. This article proposes an alternative cumulative hierarchy which does not have this drawback and provides examples of how it can be used to prove new theorems in .