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 ${\displaystyle L_{A}}$ consists of one binary relation ${\displaystyle \in }$ and one constant ${\displaystyle V}$ (Ackermann used a predicate ${\displaystyle M}$ instead). We will write ${\displaystyle x\in y}$ for ${\displaystyle \in (x,y)}$. The intended interpretation of ${\displaystyle x\in y}$ is that the object ${\displaystyle x}$ is in the class ${\displaystyle y}$. The intended interpretation of ${\displaystyle V}$ 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 ${\displaystyle L_{A}}$

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

2) Class construction axiom schema: Let ${\displaystyle F(y,z_{1},\dots ,z_{n})}$ be any formula which does not contain the variable ${\displaystyle x}$ free.

${\displaystyle \forall y(F(y,z_{1},\dots ,z_{n})\rightarrow y\in V)\rightarrow \exists x\forall y(y\in x\leftrightarrow F(y,z_{1},\dots ,z_{n}))}$

3) Reflection axiom schema: Let ${\displaystyle F(y,z_{1},\dots ,z_{n})}$ be any formula which does not contain the constant symbol ${\displaystyle V}$ or the variable ${\displaystyle x}$ free. If ${\displaystyle z_{1},\dots ,z_{n}\in V}$ then

${\displaystyle \forall y(F(y,z_{1},\dots ,z_{n})\rightarrow y\in V)\rightarrow \exists x(x\in V\land \forall y(y\in x\leftrightarrow F(y,z_{1},\dots ,z_{n}))).}$

4) Completeness axioms for ${\displaystyle V}$

${\displaystyle x\in y\land y\in V\rightarrow x\in V}$ (sometimes called the axiom of heredity)
${\displaystyle x\subseteq y\land y\in V\rightarrow x\in V.}$
${\displaystyle x\in V\land \exists y(y\in x)\rightarrow \exists y(y\in x\land \lnot \exists z(z\in y\land z\in x)).}$

Relation to Zermelo–Fraenkel set theory

Let ${\displaystyle F}$ be a first-order formula in the language ${\displaystyle L_{\in }=\{\in \}}$ (so ${\displaystyle F}$ does not contain the constant ${\displaystyle V}$). Define the "restriction of ${\displaystyle F}$ to the universe of sets" (denoted ${\displaystyle F^{V}}$) to be the formula which is obtained by recursively replacing all sub-formulas of ${\displaystyle F}$ of the form ${\displaystyle \forall xG(x,y_{1}\dots ,y_{n})}$ with ${\displaystyle \forall x(x\in V\rightarrow G(x,y_{1}\dots ,y_{n}))}$ and all sub-formulas of the form ${\displaystyle \exists xG(x,y_{1}\dots ,y_{n})}$ with ${\displaystyle \exists x(x\in V\land G(x,y_{1}\dots ,y_{n}))}$.

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

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

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".