Ackermann set theory

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

1 The language
2 The axioms
3 Relation to Zermelo-Frankel set theory
4 See also
5 Bibliography

[edit] The language

Ackerman set theory is formulated in 1st order logic. The language $L A$ consists of one binary relation $\in$ and one constant $V$ . 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.

[edit] 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 constant symbol $V$ or the variable $x$ free.

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

3) Completeness axioms for $V$

$x \in y \land y \in V \rightarrow x \in V$

$x \subseteq y \land y \in V \rightarrow x \in V$

4) Reflection axiom schema: If $z_1, \dots, z_n \in V$ then

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

5) Axiom of regularity for sets:

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

[edit] Relation to Zermelo-Frankel set theory

Let $F$ be a 1st 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 $L_\in = \{\in\}$ and A proves $F V$ then ZF proves $F$

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

[edit] See also

[edit] Bibliography

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

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Categories: Mathematical Logic | Set Theory | Systems of set theory