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Ackermann set theory

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Ackermann set theory is a version of axiomatic set theory proposed by Wilhelm Ackermann in 1956.

Contents

The language

Ackermann set theory is formulated in first-order logic. The language L A consists of one binary relation ∈ and one constant V (Ackermann used a predicate M instead). We will write x ∈ y for ∈ ( x , y ) . The intended interpretation of x ∈ 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:

∀ x ∀ y ( ∀ z ( z ∈ x ↔ z ∈ y ) → x = y ) .

2) Class construction axiom schema: Let F ( y , z 1 , … , z n ) be any formula which does not contain the variable x free.

∀ y ( F ( y , z 1 , … , z n ) → y ∈ V ) → ∃ x ∀ y ( y ∈ x ↔ F ( y , z 1 , … , z n ) )

3) Reflection axiom schema: Let F ( y , z 1 , … , z n ) be any formula which does not contain the constant symbol V or the variable x free. If z 1 , … , z n ∈ V then

∀ y ( F ( y , z 1 , … , z n ) → y ∈ V ) → ∃ x ( x ∈ V ∧ ∀ y ( y ∈ x ↔ F ( y , z 1 , … , z n ) ) ) .

4) Completeness axioms for V

x ∈ y ∧ y ∈ V → x ∈ V (sometimes called the axiom of heredity) x ⊆ y ∧ y ∈ V → x ∈ V .

5) Axiom of regularity for sets:

x ∈ V ∧ ∃ y ( y ∈ x ) → ∃ y ( y ∈ x ∧ ¬ ∃ z ( z ∈ y ∧ z ∈ x ) ) .

Relation to Zermelo–Fraenkel set theory

Let F be a first-order formula in the language L ∈ = { ∈ } (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 ∀ x G ( x , y 1 … , y n ) with ∀ x ( x ∈ V → G ( x , y 1 … , y n ) ) and all sub-formulas of the form ∃ x G ( x , y 1 … , y n ) with ∃ x ( x ∈ V ∧ G ( x , y 1 … , y n ) ) .

In 1959 Azriel Levy proved that if F is a formula of L ∈ and A proves F V , then ZF proves F

In 1970 William Reinhardt proved that if F is a formula of L ∈ and ZF proves F , then A proves F V .

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".

References

Ackermann set theory Wikipedia
(Text) CC BY-SA


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