In mathematics and logic, Ackermann set theory (AST) is an axiomatic set theory proposed by Wilhelm Ackermann in 1956.

## The language

AST is formulated in first-order logic. The language $L_{\{\in ,V\))$ of AST contains one binary relation $\in$ denoting set membership and one constant $V$ denoting the class of all sets (Ackermann used a predicate $M$ instead).

## The axioms

The axioms of AST are the following:

1. extensionality
2. heredity: $(x\in y\lor x\subseteq y)\land y\in V\to x\in V$ 3. comprehension on $V$ : for any formula $\phi$ where $x$ is not free, $\exists x\forall y(y\in x\leftrightarrow y\in V\land \phi )$ 4. Ackermann's schema: for any formula $\phi$ with free variables $a_{1},\ldots ,a_{n},x$ and no occurrences of $V$ , $a_{1},\ldots ,a_{n}\in V\land \forall x(\phi \to x\in V)\to \exists y{\in }V\forall x(x\in y\leftrightarrow \phi )$ An alternative axiomatization uses the following axioms:

1. extensionality
2. heredity
3. comprehension
4. reflection: for any formula $\phi$ with free variables $a_{1},\ldots ,a_{n)$ , $a_{1},\ldots ,a_{n}{\in }V\to (\phi \leftrightarrow \phi ^{V})$ 5. regularity

$\phi ^{V)$ denotes the relativization of $\phi$ to $V$ , which replaces all quantifiers in $\phi$ of the form $\forall x$ and $\exists x$ by $\forall x{\in }V$ and $\exists x{\in }V$ , respectively.

## Relation to Zermelo–Fraenkel set theory

Let $L_{\{\in \))$ be the language of formulas that do not mention $V$ .

In 1959, Azriel Levy proved that if $\phi$ is a formula of $L_{\{\in \))$ and AST proves $\phi ^{V)$ , then ZF proves $\phi$ .

In 1970, William N. Reinhardt proved that if $\phi$ is a formula of $L_{\{\in \))$ and ZF proves $\phi$ , then AST proves $\phi ^{V)$ .

Therefore, AST and ZF are mutually interpretable in conservative extensions of each other. Thus they are equiconsistent.

A remarkable feature of AST is that, unlike NBG and its variants, a proper class can be an element of another proper class.

## AST and category theory

An extension of AST called ARC was developed by F.A. Muller, who 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".