Another approach is taken by the von NeumannโBernaysโGรถdel axioms (NBG);
classes are the basic objects in this theory, and a set is then defined to be a class
that is an element of some other class.
However, the class existence axioms of NBG are restricted so that
they only quantify over sets, rather than over all classes.
This causes NBG to be a conservative extension of ZF.
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