Ackermann set theory explained

In mathematics and logic, Ackermann set theory (AST, also known as

A ^{*/V} ^[1]) is an axiomatic set theory proposed by Wilhelm Ackermann in 1956.^[2]

AST differs from Zermelo–Fraenkel set theory (ZF) in that it allows proper classes, that is, objects that are not sets, including a class of all sets.It replaces several of the standard ZF axioms for constructing new sets with a principle known as Ackermann's schema. Intuitively, the schema allows a new set to be constructed if it can be defined by a formula which does not refer to the class of all sets.In its use of classes, AST differs from other alternative set theories such as Morse–Kelley set theory and Von Neumann–Bernays–Gödel set theory in that a class may be an element of another class.

William N. Reinhardt established in 1970 that AST is effectively equivalent in strength to ZF, putting it on equal foundations. In particular, AST is consistent if and only if ZF is consistent.

Preliminaries

L _{\{\in,V\} } of AST contains one binary relation \in denoting set membership and one constant V denoting the class of all sets. Ackermann used a predicate M instead of V ; this is equivalent as each of M and V can be defined in terms of the other.^[3]

We will refer to elements of

V as sets, and general objects as classes. A class that is not a set is called a proper class.

Axioms

The following formulation is due to Reinhardt.^[4] The five axioms include two axiom schemas.Ackermann's original formulation included only the first four of these, omitting the axiom of regularity.^{[5] [6] [7]}

1. Axiom of extensionality

If two classes have the same elements, then they are equal.

\forall x (x \in A \leftrightarrow x \in B) \to A = B.

This axiom is identical to the axiom of extensionality found in many other set theories, including ZF.

2. Heredity

Any element or a subset of a set is a set.

(x \in y \lor x \subseteq y) \land y \in V \to x \in V.

3. Comprehension schema

For any property, we can form the class of sets satisfying that property. Formally, for any formula

\phi where X is not free:\exists X \forall x (x \in X \leftrightarrow x \in V \land \phi).

That is, the only restriction is that comprehension is restricted to objects in

V . But the resulting object is not necessarily a set.

4. Ackermann's schema

For any formula

\phi with free variables a _{1,} \ldots, a _{n,} x and no occurrences of V :a _{1,} \ldots, a _{n} \in V \land \forall x (\phi \to x \in V) \to \exists X {\in} V \forall x (x \in X \leftrightarrow \phi).

Ackermann's schema is a form of set comprehension that is unique to AST. It allows constructing a new set (not just a class) as long as we can define it by a property that does not refer to the symbol

V . This is the principle that replaces ZF axioms such as pairing, union, and power set.

5. Regularity

Any non-empty set contains an element disjoint from itself:

\forall x \in V (x = \varnothing \lor \exists y (y \in x \land y \cap x = \varnothing)).

Here,

y \cap x = \varnothing is shorthand for \not \exists z (z \in x \land z \in y) . This axiom is identical to the axiom of regularity in ZF.

This axiom is conservative in the sense that without it, we can simply use comprehension (axiom schema 3) to restrict our attention to the subclass of sets that are regular.^[4]

Alternative formulations

Ackermann's original axioms did not include regularity, and used a predicate symbol

M instead of the constant symbol V .^[2] We follow Lévy and Reinhardt in replacing instances of Mx with x \in V . This is equivalent because M can be given a definition as x \in V , and conversely, the set V can be obtained in Ackermann's original formulation by applying comprehension to the predicate \phi = True .^[3]

In axiomatic set theory, Ralf Schindler replaces Ackermann's schema (axiom schema 4) with the following reflection principle:for any formula

\phi with free variables a _{1,} \ldots, a _{n} ,a _{1,} \ldots, a _{n} {\in} V \to (\phi \leftrightarrow \phi ^{V).}

Here,

\phi ^{V} denotes the relativization of \phi to V , which replaces all quantifiers in \phi of the form \forall x and \exists x by \forall x {\in} V and \exists x {\in} V , respectively.^[8]

Relation to Zermelo–Fraenkel set theory

Let

L _{\{\in\} } be the language of formulas that do not mention V .

In 1959, Azriel Lévy proved that if

\phi is a formula of L _{\{\in\} } and AST proves \phi ^{V} , then ZF proves \phi .^[3]

In 1970, William N. Reinhardt proved that if

\phi is a formula of L _{\{\in\} } and ZF proves \phi , then AST proves \phi ^{V} .^[4]

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.^[7]

Extensions

An extension of AST for category theory called ARC was developed by F.A. Muller. Muller 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".^[9]

See also

• Foundations of mathematics
• List of alternative set theories
• Zermelo set theory

Notes and References

1. A. Lévy, A hierarchy of formulas in set theory (1974), p.69. Memoirs of the Americal Mathematical Society no. 57
2. Ackermann . Wilhelm . Wilhelm Ackermann . August 1956 . Zur Axiomatik der Mengenlehre . Mathematische Annalen . 131 . 4 . 336–345 . 10.1007/BF01350103 . 120876778 . 9 September 2022.
3. Lévy . Azriel . Azriel Lévy . June 1959 . On Ackermann's Set Theory . The Journal of Symbolic Logic . 24 . 2 . 154–166 . 10.2307/2964757 . 2964757 . 31382168 . 9 September 2022.
4. Reinhardt . William N. . October 1970 . Ackermann's set theory equals ZF . Annals of Mathematical Logic . 2 . 2 . 189–249 . 10.1016/0003-4843(70)90011-2 .
5. Kanamori . Akihiro . Akihiro Kanamori . July 2006 . Levy and set theory . Annals of Pure and Applied Logic . 140 . 1 . 233–252 . 10.1016/j.apal.2005.09.009 .
6. Web site: Alternative Axiomatic Set Theories . Holmes . M. Randall . Sep 21, 2021 . Stanford Encyclopedia of Philosophy . Metaphysics Research Lab, Stanford University . 8 September 2022.
7. Book: Fraenkel . Abraham A. . Abraham Fraenkel . Bar-Hillel . Yehoshua . Yehoshua Bar-Hillel . Levy . Azriel . Azriel Levy . December 1, 1973 . Foundations of Set Theory . 9780080887050 . Studies in Logic and the Foundations of Mathematics . 67 . 7.7. The System of Ackermann . 148–153.
8. Book: Schindler, Ralf . 23 May 2014 . Set Theory: Exploring Independence and Truth . Springer, Cham . 20–21 . 978-3-319-06724-7 . Chapter 2: Axiomatic Set Theory. 10.1007/978-3-319-06725-4_2 .
9. Muller . F. A. . Sep 2001 . Sets, Classes, and Categories . The British Journal for the Philosophy of Science . 52 . 3 . 539–573 . 10.1093/bjps/52.3.539 . 3541928 . 9 September 2022.