In mathematics and logic, Ackermann set theory (AST) is an axiomatic set theory proposed by Wilhelm Ackermann in 1956.[1]

The language

AST is formulated in first-order logic. The language of AST contains one binary relation denoting set membership and one constant denoting the class of all sets (Ackermann used a predicate instead).

The axioms

The axioms of AST are the following:[2][3][4]

  1. extensionality
  2. heredity:
  3. comprehension on : for any formula where is not free,
  4. Ackermann's schema: for any formula with free variables and no occurrences of ,

An alternative axiomatization uses the following axioms:[5]

  1. extensionality
  2. heredity
  3. comprehension
  4. reflection: for any formula with free variables ,
  5. regularity

denotes the relativization of to , which replaces all quantifiers in of the form and by and , respectively.

Relation to Zermelo–Fraenkel set theory

Let be the language of formulas that do not mention .

In 1959, Azriel Levy proved that if is a formula of and AST proves , then ZF proves .[6]

In 1970, William N. Reinhardt proved that if is a formula of and ZF proves , then AST proves .[7]

Therefore, AST and ZF are mutually interpretable in conservative extensions of each other. Thus they are equiconsistent.

A remarkable feature of AST is that, unlike NBG and its variants, a proper class can be an element of another proper class.[4]

AST and category theory

An extension of AST called ARC was developed by F.A. Muller, 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".[8]

See also


  1. ^ Ackermann, Wilhelm (August 1956). "Zur Axiomatik der Mengenlehre". Mathematische Annalen. 131 (4): 336–345. doi:10.1007/BF01350103. S2CID 120876778. Retrieved 9 September 2022.
  2. ^ Kanamori, Akihiro (July 2006). "Levy and set theory". Annals of Pure and Applied Logic. 140 (1): 233–252. doi:10.1016/j.apal.2005.09.009.
  3. ^ Holmes, M. Randall (Sep 21, 2021). "Alternative Axiomatic Set Theories". Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 8 September 2022.
  4. ^ a b Fraenkel, Abraham A.; Bar-Hillel, Yehoshua; Levy, Azriel (December 1, 1973). "7.7. The System of Ackermann". Foundations of Set Theory. Studies in Logic and the Foundations of Mathematics. Vol. 67. pp. 148–153. ISBN 9780080887050.
  5. ^ Schindler, Ralf (23 May 2014). "Chapter 2: Axiomatic Set Theory". Set Theory: Exploring Independence and Truth. Springer, Cham. pp. 20–21. doi:10.1007/978-3-319-06725-4_2. ISBN 978-3-319-06724-7.
  6. ^ Lévy, Azriel (June 1959). "On Ackermann's Set Theory". The Journal of Symbolic Logic. 24 (2): 154–166. doi:10.2307/2964757. JSTOR 2964757. S2CID 31382168. Retrieved 9 September 2022.
  7. ^ Reinhardt, William N. (October 1970). "Ackermann's set theory equals ZF". Annals of Mathematical Logic. 2 (2): 189–249. doi:10.1016/0003-4843(70)90011-2.
  8. ^ Muller, F. A. (Sep 2001). "Sets, Classes, and Categories". The British Journal for the Philosophy of Science. 52 (3): 539–573. doi:10.1093/bjps/52.3.539. JSTOR 3541928. Retrieved 9 September 2022.