Ackermann set theory
Encyclopedia
Ackermann set theory is a version of axiomatic set theory proposed by Wilhelm Ackermann
in 1956.
. 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.
1) Axiom of extensionality
:
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
5) Axiom of regularity for sets
:
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 
.
Wilhelm Ackermann
Wilhelm Friedrich Ackermann was a German mathematician best known for the Ackermann function, an important example in the theory of computation....
in 1956.
The language
Ackermann set theory is formulated in first-order logicFirst-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate 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 language1) Axiom of extensionality
Axiom of extensionality
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory.- Formal statement :...
:
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 then4) Completeness axioms for
5) Axiom of regularity for sets
Axiom of regularity
In mathematics, the axiom of regularity is one of the axioms of Zermelo-Fraenkel set theory and was introduced by...
:
Relation to Zermelo–Fraenkel set theory
Let be a first-order formulaFirst-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
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
Azriel Levy
Azriel Levy is an Israeli mathematician, logician, and a professor emeritus at the Hebrew University of Jerusalem....
proved that if
is a formula of and A proves , then ZFZermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...
proves
In 1970 William Reinhardt proved that if
is a formula of and ZF proves , then A proves .