- Ackermann set theory
Ackermann set theory is a version of
axiomatic set theory proposed byWilhelm Ackermann in 1956.The language
Ackermann set theory is formulated in
first-order logic . The language L_A consists of one binary relation in and one constant V (Ackermann used a predicate M instead). We will write x in y for in(x,y). The intended interpretation of x in y is that the object x is in the class y. The intended interpretation of V 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 L_A
1)
Axiom of extensionality ::forall x forall y ( forall z (z in x leftrightarrow z in y) ightarrow x = y)2) Class construction axiom schema: Let F(y,z_1, dots, z_n) be any formula which does not contain the variable x free. :forall y (F(y, z_1, dots, z_n) ightarrow y in V) ightarrow exists x forall y (y in x leftrightarrow F(y,z_1, dots, z_n) )
3) Reflection axiom schema: Let F(y,z_1, dots, z_n) be any formula which does not contain the constant symbol V or the variable x free. If z_1, dots, z_n in V then:forall y (F(y, z_1, dots, z_n) ightarrow y in V) ightarrow exists x (x in V land forall y (y in x leftrightarrow F(y, z_1, dots, z_n)))
4) Completeness axioms for V:x in y land y in V ightarrow x in V:x subseteq y land y in V ightarrow x in V
5) Axiom of regularity for sets::x in V land exists y ( y in x) ightarrow exists y ( y in x land lnot exists z (z in y land z in x))
Relation to Zermelo-Frankel set theory
Let F be a first-order formula in the language L_in = {in} (so F does not contain the constant V). Define the "restriction of F to the universe of sets" (denoted F^V) to be the formula which is obtained by recursively replacing all
sub-formulas of F of the form forall x G(x,y_1dots, y_n) with forall x (x in V ightarrow G(x,y_1dots, y_n)) and all sub-formulas of the form exists x G(x,y_1dots, y_n) with exists x (x in V land G(x,y_1dots, y_n)).In 1959
Azriel Levy proved that if F is a formula of L_in and A proves F^V, then ZF proves FIn 1970
William Reinhardt proved that if F is a formula of L_in and ZF proves F, then A proves F^V.See also
*
Axiomatic set theory
*Zermelo set theory Bibliography
* 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"
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