Ackermann set theory

In mathematics and logic, Ackermann set theory (AST, also known as ${\displaystyle A^{*}/V}$^[1]) is an axiomatic set theory proposed by Wilhelm Ackermann in 1956.^[2]

AST differs from Zermelo–Fraenkel set theory (ZF) in that it allows proper classes, that is, objects that are not sets, including a class of all sets. It replaces several of the standard ZF axioms for constructing new sets with a principle known as Ackermann's schema. Intuitively, the schema allows a new set to be constructed if it can be defined by a formula which does not refer to the class of all sets. In its use of classes, AST differs from other alternative set theories such as Morse–Kelley set theory and Von Neumann–Bernays–Gödel set theory in that a class may be an element of another class.

William N. Reinhardt established in 1970 that AST is effectively equivalent in strength to ZF, putting it on equal foundations. In particular, AST is consistent if and only if ZF is consistent.