Definition:Gödel-Bernays Axioms

Definition

The Gödel-Bernays axioms are a conservative extension of the Zermelo-Fraenkel axioms with the axiom of choice (ZFC) that allow comprehension of classes.

Although not the standard axioms of set theory, particularly in category theory they spare us any set of all sets-type paradoxes.

Axioms for Sets

The first five axioms are identical to the axioms of the same names from ZFC.

The quantified variables range over the universe of sets.

The Axiom of Extension

Two sets are equal if and only if they contain the same elements:

$\forall x: \left({x \in A \iff x \in B}\right) \iff A = B$

The order of the elements in the sets is immaterial.

The Axiom of Pairing

For any two sets, there exists a set to which only those two sets are elements:

$\forall A: \forall B: \exists x: \forall y: \left({y \in x \iff y = A \lor y = B}\right)$

Thus it is possible to create a set containing any two sets that you have already created.

Otherwise known as the Axiom of the Unordered Pair.

The Axiom of Unions

For every set of sets, there exists a set (the sum or union set) that contains all the elements (and only those elements) that belong to at least one of the sets in the set:

$\forall A: \exists x: \forall y: \left({y \in x \iff \exists z: \left({z \in A \land y \in z}\right)}\right)$

The Axiom of Powers

For every set, there exists a set of sets that contains amongst its elements all the subsets of the given set.

$\forall x: \exists y: \left({\forall z: \left({z \in y \iff \left({w \in z \implies w \in x}\right)}\right)}\right)$

The Axiom of Infinity

There exists a set containing:

$(1): \quad$ a set with no elements
$(2): \quad$ the successor of each of its elements.

That is:

$\exists x: \left({\left({\exists y: y \in x \land \forall z: \neg \left({z \in y}\right)}\right) \land \forall u: u \in x \implies u^+ \in x}\right)$

Axioms for Classes

In the remaining axioms, the quantified variables range over classes.

The first two differ from the ZFC axioms with the same names in this way only.

The last two have no analogue among the ZFC axioms.

The Axiom of Extension

Two classes are equal if they contain the same elements:

$\forall x: \left({x \in A \iff x \in B}\right) \iff A = B$

The order of the elements in the sets is immaterial.

The Axiom of Foundation

For any non-empty class, there is an element of the class that shares no element with the class.

$\forall S: \neg \left({S = z: \forall y: \left({\neg \left({y \in z}\right)}\right)}\right) \implies \exists x \in S: \neg \left({\exists w: w \in S \land w \in x}\right)$

Class Comprehension

For any formula $\phi$ containing no quantifiers over classes, there is a class $A$ such that:

$\forall x: \left({x \in A \iff \phi \left({x}\right)}\right)$

The Axiom of Limitation of Size

For any class $\mathcal C$, a set $x$ such that $x = \mathcal C$ exists if and only if there is no bijection between $\mathcal C$ and the universe.