Axiomatic set theory proposed by Wilhelm Ackermann

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

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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 $L_{A))$

1) Axiom of extensionality:

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

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 $V$

- $x\in y\land y\in V\rightarrow x\in V$ (sometimes called the axiom of heredity)
- $x\subseteq y\land y\in V\rightarrow x\in V.$

5) Axiom of regularity for sets:

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

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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 $\forall xG(x,y_{1}\dots ,y_{n})$ with $\forall x(x\in V\rightarrow G(x,y_{1}\dots ,y_{n}))$ and all sub-formulas of the form $\exists xG(x,y_{1}\dots ,y_{n})$ 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 $F^{V))$.

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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".