Ackermann set theory
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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 consists of one binary relation and one constant (Ackermann used a predicate instead). We will write for . The intended interpretation of is that the object is in the class . The intended interpretation of 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
2) Class construction axiom schema: Let be any formula which does not contain the variable free.
3) Reflection axiom schema: Let be any formula which does not contain the constant symbol or the variable free. If then
4) Completeness axioms for
- (sometimes called the axiom of heredity)
5) Axiom of regularity for sets:
Relation to Zermelo–Fraenkel set theory[]
Let be a first-order formula in the language (so does not contain the constant ). Define the "restriction of to the universe of sets" (denoted ) to be the formula which is obtained by recursively replacing all of of the form with and all sub-formulas of the form with .
In 1959 Azriel Levy proved that if is a formula of and A proves , then ZF proves
In 1970 proved that if is a formula of and ZF proves , then A proves .
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".
See also[]
References[]
- 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
- , "Ackermann's set theory equals ZF" Annals of Mathematical Logic Vol. 2, 1970 no. 2, 189--249
- A.A.Fraenkel, Y. Bar-Hillel, A.Levy, 1973. Foundations of Set Theory, second edition, North-Holand, 1973.
- F.A. Muller, "Sets, Classes, and Categories" British Journal for the Philosophy of Science 52 (2001) 539-573.
- Systems of set theory