Ackermann set theory

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a> and <a href="/facts/Logic/8JnpJLtt">logic</a>, Ackermann set theory (AST, also known as 
 
 
 
 
 A
 
 ∗
 
 
 
 /
 
 V
 
 
 {\displaystyle A^{*}/V}
 
) is an axiomatic <a href="/facts/Set_theory/1ptfsvUP">set theory</a> proposed by <a href="/facts/Wilhelm_Ackermann/8bPlUHSE">Wilhelm Ackermann</a> in 1956.
AST differs from <a href="/facts/Zermelo%E2%80%93Fraenkel_set_theory/UYIsPbXw">Zermelo–Fraenkel set theory</a> (ZF) in that it allows <a href="/facts/Class_(set_theory)/F5EmDd9b">proper classes</a>, 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 <a href="/facts/Morse%E2%80%93Kelley_set_theory/P4hoeLYC">Morse–Kelley set theory</a> and <a href="/facts/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory/5wLhAIMz">Von Neumann–Bernays–Gödel set theory</a> 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 <a href="/facts/Consistency/IHkdnA9M">consistent</a> if and only if ZF is consistent.

Ackermann set theory open-in-new

Ackermann set theory