Ackermann set theory

Ackermann set theory is a version of axiomatic set theory proposed by Wilhelm Ackermann in 1956.

The language

Ackermann set theory is formulated in first-order logic. The language $L_A$ consists of one binary relation $\in$ and one constant $V$ (Ackermann used a predicate $M$ instead). We will write $x \in y$ for $\in(x,y)$. The intended interpretation of $x \in y$ is that the object $x$ is in the class $y$. The intended interpretation of $V$ is the class of all sets.

The axioms

The axioms of Ackermann set theory, collectively referred to as A, consists of the universal closure of the following formulas in the language $L_A$

$\forall x \forall y ( \forall z (z \in x \leftrightarrow z \in y) \rightarrow x = y).$

2) Class construction axiom schema: Let $F(y,z_1, \dots, z_n)$ be any formula which does not contain the variable $x$ free.

$\forall y (F(y, z_1, \dots, z_n) \rightarrow y \in V) \rightarrow \exists x \forall y (y \in x \leftrightarrow F(y,z_1, \dots, z_n) )$

3) Reflection axiom schema: Let $F(y,z_1, \dots, z_n)$ be any formula which does not contain the constant symbol $V$ or the variable $x$ free. If $z_1, \dots, z_n \in V$ then

$\forall y (F(y, z_1, \dots, z_n) \rightarrow y \in V) \rightarrow \exists x (x \in V \land \forall y (y \in x \leftrightarrow F(y, z_1, \dots, z_n))).$

4) Completeness axioms for $V$

$x \in y \land y \in V \rightarrow x \in V$
$x \subseteq y \land y \in V \rightarrow x \in V.$
$x \in V \land \exists y ( y \in x) \rightarrow \exists y ( y \in x \land \lnot \exists z (z \in y \land z \in x)).$

Relation to Zermelo–Fraenkel set theory

Let $F$ be a first-order formula in the language $L_\in = \{\in\}$ (so $F$ does not contain the constant $V$). Define the "restriction of $F$ to the universe of sets" (denoted $F^V$) to be the formula which is obtained by recursively replacing all sub-formulas of $F$ of the form $\forall x G(x,y_1\dots, y_n)$ with $\forall x (x \in V \rightarrow G(x,y_1\dots, y_n))$ and all sub-formulas of the form $\exists x G(x,y_1\dots, y_n)$ with $\exists x (x \in V \land G(x,y_1\dots, y_n))$.

In 1959 Azriel Levy proved that if $F$ is a formula of $L_\in$ and A proves $F^V$, then ZF proves $F$

In 1970 William Reinhardt proved that if $F$ is a formula of $L_\in$ and ZF proves $F$, then A proves $F^V$.

Ackermann set theory and Category theory

The most remarkable feature of Ackermann set theory is that, unlike Von Neumann–Bernays–Gödel set theory, a proper class can be an element of another proper class (see Fraenkel, Bar-Hillel, Levy(1973), p. 153).

An extension (named ARC) of Ackermann set theory was developed by F.A. Muller(2001), 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".