Definition:Gödel-Bernays Axioms

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.

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.