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 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 sub-formulas 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 William Reinhardt 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
- Reinhardt, William, "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.