Class (set theory)

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In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context. In work on ZF set theory, the notion of class is informal, whereas other set theories, such as NBG set theory, axiomatize the notion of "class".

Every set is a class, no matter which foundation is chosen. A class that is not a set is (informally in Zermelo–Fraenkel) called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems.

Various important concepts in mathematics are commonly described with classes. Examples include large categories and the class-field of surreal numbers.

In ZF set theory, classes exist only in the metalanguage, as equivalence classes of logical formulas. The axioms of ZF do not 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; 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. In other, less standard 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.

The paradoxes of naive set theory can be explained in terms of the inconsistent assumption that "all classes are sets". With a rigorous foundation, these paradoxes instead suggest proofs that certain classes are proper. For example, Russell's paradox suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox suggests that the class of all ordinal numbers is proper. One way to prove that a class is proper is to place it in bijection with the class of ordinals; see, for instance, the proof that there is no free complete lattice.

The word "class" was sometimes used synonymously with "set". This usage dates from a historical period where classes and sets were not distinguished as they are in modern terminology. Many discussions of "classes" in the 19th century and earlier are really referring to sets, or perhaps to a more ambiguous concept.