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 ${\displaystyle \in (x,y)}$. 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 $L_{A}$

1) Axiom of extensionality:

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

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

${\displaystyle x\in y\land y\in V\rightarrow x\in V}$ (sometimes called the axiom of heredity)

${\displaystyle x\subseteq y\land y\in V\rightarrow x\in V.}$

5) Axiom of regularity for sets:

${\displaystyle x\in V\land \exists y(y\in x)\rightarrow \exists y(y\in x\land \lnot \exists z(z\in y\land z\in x)).}$

Relation to Zermelo–Fraenkel set theory

Let $F$ be a first-order formula in the language ${\displaystyle L_{\in }=\{\in \}}$ (so $F$ does not contain the constant $V$). Define the "restriction of $F$ to the universe of sets" (denoted ${\displaystyle F^{V}}$) to be the formula which is obtained by recursively replacing all sub-formulas of $F$ 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 $F$ is a formula of ${\displaystyle L_{\in }}$ and A proves ${\displaystyle F^{V}}$, then ZF proves $F$

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

See also

• Zermelo set theory

References

• Ackermann, Wilhelm "Zur Axiomatik der Mengenlehre" in Mathematische Annalen, 1956, Vol. 131, pp. 336--345.
• Levy, Azriel, "On Ackermann's set theory" Journal of Symbolic Logic Vol. 24, 1959 154--166
• Reinhardt, William, "Ackermann's set theory equals ZF" Annals of Mathematical Logic Vol. 2, 1970 no. 2, 189--249
• A.A.Fraenkel, Y. Bar-Hillel, A.Levy, 1973. Foundations of Set Theory, second edition, North-Holand, 1973.
• F.A. Muller, "Sets, Classes, and Categories" British Journal for the Philosophy of Science 52 (2001) 539-573.