Ackermann set theory

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

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[edit] The language

Ackerman set theory is formulated in 1st order logic. The language LA 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 LA

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 FV) 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 FV 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 FV.

[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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