Ackermann set theory is a version of axiomatic set theory proposed by Wilhelm Ackermann in 1956.
Contents
The language
Ackermann set theory is formulated in first-order logic. The language
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
1) Axiom of extensionality:
2) Class construction axiom schema: Let
3) Reflection axiom schema: Let
4) Completeness axioms for
5) Axiom of regularity for sets:
Relation to Zermelo–Fraenkel set theory
Let
In 1959 Azriel Levy proved that if
In 1970 William Reinhardt proved that if
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".