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 axioms of Ackermann set theory, collectively referred to as A, consists of the universal closure of  the following formulas in the language formula_1 1) Axiom of extensionality: 2) Class construction axiom schema: Let formula_13 be any formula which does not contain the variable formula_8 free. 3) Reflection axiom schema: Let formula_13 be any formula which does not contain the constant symbol formula_3 or the variable formula_8 free. If formula_19 then 4) Completeness axioms for formula_3 5) Axiom of regularity for sets: Relation to Zermelo–Fraenkel set theory. Let formula_25 be a first-order formula in the language formula_26 (so formula_25 does not contain the constant formula_3). Define the "restriction of formula_25 to the universe of sets  (denoted formula_30) to be the formula which is obtained by recursively replacing all sub-formulas of formula_25 of the form formula_32 with formula_33 and all sub-formulas of the form formula_34 with formula_35. In 1959 Azriel Levy proved that if formula_25 is a formula of formula_37 and A proves formula_30, 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 	Ackermann set theory
16604230	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".