Ackermann set theory

The Ackermann set theory is an axiomatic set theory, which was given in 1955 by Wilhelm Ackermann. He tried in it to implement Cantor set definition in a precise system of axioms.

The Ackermann set theory extends the Zermelo -Fraenkel set theory ZFC to classes (there: populations ), but differs from the well-known Neumann - Bernays - Gödel set theory by the fact that proper classes can also be elements of other classes and therefore even small real classes are. The ZFC axioms are there in a real portion that meets the foundation axiom (you can discard it with Neumann cumulative hierarchy). Therefore, the Ackermann set theory includes an extended amount of area with non- well-founded sets and can be viewed as a generalization of the usual ZFC set theory and the Zermelo set theory.

The Ackermann axioms

Ackermann's axiom system is based on the first order predicate logic with identity, the two -digit code and the relation -digit predicate and each has an axiom schema, and an axiom for classes and for amounts:

  • Class comprehension: classes of sets are existent:
  • Class extensionality: Classes with the same elements are the same:
  • Quantity of comprehension: Exclusively occupied with quantities classes of sets are sets:
  • Elements and sub-classes of sets are sets:


The axiom of choice is not included in the above list. By adding the Ackermann set theory is obtained with the axiom of choice, which he added in a formulation with Hilbert's choice operator.

Ackermann formulated in his work, a version of his axiom system, the primary elements taken into account, as well as an ajar to the type theory version.