Class (set theory)

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", e.g., as entities that are not members of another entity.

Every set is a class, no matter which foundation is chosen. A class that is not a set (informally in Zermelo–Fraenkel) is 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.

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

Examples

The collection of all algebraic objects of a given type will usually be a proper class. Examples include the class of all groups, the class of all vector spaces, and many others. In category theory, a category whose collection of objects forms a proper class (or whose collection of morphisms forms a proper class) is called a large category.

The surreal numbers are a proper class of objects that has the properties of a field.

Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets, the class of all ordinal numbers, and the class of all cardinal numbers.

One way to prove that a class is proper is to place it in bijection with the class of all ordinal numbers. This method is used, for example, in the proof that there is no free complete lattice.

Paradoxes

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.

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 $\mathcal A$ is a structure interpreting ZF, then the metalanguage expression $\{x\mid x=x \}$ is interpreted in $\mathcal A$ by the collection of all the elements from the domain of $\mathcal A$; that is, all the sets in $\mathcal A$. 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.

References

• Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics (third millennium ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-44085-7 
• Levy, A. (1979), Basic Set Theory, Berlin, New York: Springer-Verlag