Classes in Formal Set Theories
ZF set theory does not formalize the notion of classes. They can instead be described in the metalanguage, as equivalence classes of logical formulas. For example, if is a structure interpreting ZF, then the metalanguage expression is interpreted in by the collection of all the elements from the domain of ; that is, all the sets in . So we can identify the "class of all sets" with the predicate x=x or any equivalent predicate.
Because classes do not have any formal status in the theory of ZF, the axioms of ZF do not immediately apply to classes. However, if an inaccessible cardinal κ is assumed, then the sets of smaller rank form a model of ZF (a Grothendieck universe), and its subsets can be thought of as "classes".
Another approach is taken by the von Neumann–Bernays–Gödel 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 set 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.
Morse–Kelley set theory admits proper classes as basic objects, like NBG, but also allows quantification over all proper classes in its set existence axioms. This causes MK to be strictly stronger than both NBG and ZF.
In other set theories, such as New Foundations or the theory of semisets, the concept of "proper class" still makes sense (not all classes are sets) but the criterion of sethood is not closed under subsets. For example, any set theory with a universal set has proper classes which are subclasses of sets.
Read more about this topic: Class (set Theory)
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