Ackermann set theory [SEP] named after
Ackermann set theory
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 formula_1 consists of one binary relation formula_2 and one constant formula_3 (Ackermann used a predicate formula_4 instead). We will write formula_5 for formula_6. The intended interpretation of formula_5 is that the object formula_8 is in the class formula_9. The intended interpretation of formula_3 is the class of all sets.
The axioms.
The
, then ZF proves formula_25
In 1970 William Reinhardt proved that if formula_25 is a formula of formula_37 and ZF proves formula_25, then A proves formula_30.
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.