Current differencing transconductance amplifier

Template:More footnotes 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 $L_A$

1) Axiom of extensionality:

$\mathrm{\forall }x\mathrm{\forall }y(\mathrm{\forall }z(z\in x\leftrightarrow z\in y)\to x=y).$

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

$\mathrm{\forall }y(F(y,z_1,\dots ,z_n)\to y\in V)\to \mathrm{\exists }x\mathrm{\forall }y(y\in x\leftrightarrow F(y,z_1,\dots ,z_n))$

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

$\mathrm{\forall }y(F(y,z_1,\dots ,z_n)\to y\in V)\to \mathrm{\exists }x(x\in V\wedge \mathrm{\forall }y(y\in x\leftrightarrow F(y,z_1,\dots ,z_n))).$

4) Completeness axioms for $V$

$x\in y\wedge y\in V\to x\in V$

$x\subseteq y\wedge y\in V\to x\in V.$

5) Axiom of regularity for sets:

$x\in V\wedge \mathrm{\exists }y(y\in x)\to \mathrm{\exists }y(y\in x\wedge \mathrm{\neg }\mathrm{\exists }z(z\in y\wedge z\in x)).$

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 $\mathrm{\forall }xG(x,y_1\dots ,y_n)$ with $\mathrm{\forall }x(x\in V\to G(x,y_1\dots ,y_n))$ and all sub-formulas of the form $\mathrm{\exists }xG(x,y_1\dots ,y_n)$ with $\mathrm{\exists }x(x\in V\wedge G(x,y_1\dots ,y_n))$.

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".

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.