In mathematics and logic, Ackermann set theory (AST, also known as
[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.
AST is formulated in first-order logic.
of AST contains one binary relation
denoting set membership and one constant
denoting the class of all sets.
as sets, and general objects as classes.
The following formulation is due to Reinhardt.
Ackermann's original formulation included only the first four of these, omitting the axiom of regularity.
[5][6][7][note 1] If two classes have the same elements, then they are equal.
This axiom is identical to the axiom of extensionality found in many other set theories, including ZF.
But the resulting object is not necessarily a set.
: Ackermann's schema is a form of set comprehension that is unique to AST.
It allows constructing a new set (not just a class) as long as we can define it by a property that does not refer to the symbol
This is the principle that replaces ZF axioms such as pairing, union, and power set.
Any non-empty set contains an element disjoint from itself: Here,
This axiom is conservative in the sense that without it, we can simply use comprehension (axiom schema 3) to restrict our attention to the subclass of sets that are regular.
[4] Ackermann's original axioms did not include regularity, and used a predicate symbol
[2] We follow Lévy and Reinhardt in replacing instances of
can be obtained in Ackermann's original formulation by applying comprehension to the predicate
[3] In axiomatic set theory, Ralf Schindler replaces Ackermann's schema (axiom schema 4) with the following reflection principle: for any formula
[4] Therefore, AST and ZF are mutually interpretable in conservative extensions of each other.
A remarkable feature of AST is that, unlike NBG and its variants, a proper class can be an element of another proper class.
[7] An extension of AST for category theory called ARC was developed by F.A.
Muller stated that ARC "founds Cantorian set-theory as well as category-theory and therefore can pass as a founding theory of the whole of mathematics".