Class (set theory)
A collection of objects defined by a property, often to avoid set-theoretic paradoxes.
In set theory and related mathematics, a class is a collection of objects (typically sets) defined by a shared property. Classes function like set-like collections but are distinct from sets to prevent paradoxes, most notably Russell's paradox. The exact meaning of "class" depends on the foundational system. In Zermelo–Fraenkel set theory, the concept is informal, while in von Neumann–Bernays–Gödel set theory, the idea of a "proper class" is formally defined—for instance, as an entity that cannot be a member of any other entity.
A class that is not a set is called a proper class (informally in Zermelo–Fraenkel), while a class that is a set is sometimes called a small class. For example, the class of all ordinal numbers and the class of all sets are proper classes in many formal systems. In Quine's work on set theory, the term "ultimate class" is often used instead of "proper class," highlighting that in his systems certain classes cannot be members and thus are the final link in any membership chain.
Outside set theory, "class" is sometimes used interchangeably with "set." This usage comes from a time before the modern distinction between classes and sets was made. Many 19th-century discussions of "classes" actually refer to sets, or simply do not consider that some classes might fail to be sets.
**Examples** The collection of all algebraic structures of a given type is usually a proper class—for instance, the class of all groups, the class of all vector spaces, and many others. In category theory, a category whose collection of objects (or morphisms) forms a proper class is called a large category. The surreal numbers form a proper class with the properties of a field. Within set theory, many collections turn out to be proper classes, such as the class of all sets (the universal class), the class of all ordinal numbers, and the class of all cardinal numbers. One way to prove a class is proper is to put it in bijection with the class of all ordinal numbers; this method is used, for example, to show there is no free complete lattice on three or more generators.
**Paradoxes** The paradoxes of naive set theory arise from the inconsistent assumption that all classes are sets. With a rigorous foundation, these paradoxes instead show that certain classes are proper (i.e., not sets). For instance, Russell's paradox proves that the class of all sets that do
- definition
- A collection of mathematical objects defined by a shared property
- key distinction
- A class that is not a set is called a proper class; a class that is a set is sometimes called a small class
- examples of proper classes
- Class of all ordinal numbers, class of all sets, class of all groups, class of all vector spaces, surreal numbers
- role in paradoxes
- Russell's paradox suggests the class of all sets that do not contain themselves is proper; Burali-Forti paradox suggests the class of all ordinal numbers is proper
- treatment in ZF
- Informal; formulas with classes must be reduced syntactically to formulas without classes
- treatment in NBG
- Classes are basic objects; a set is a class that is an element of some other class
- alternative terminology
- In Quine's set-theoretical writing, 'ultimate class' is used instead of 'proper class'
Lore & Background
The concept of class arose to handle collections that are too large or problematic to be sets, such as the class of all sets or the class of all ordinal numbers. In naive set theory, the tacit assumption that all classes are sets leads to paradoxes like Russell's paradox; with a rigorous foundation, these paradoxes instead prove that certain classes are proper (i.e., not sets). Outside set theory, the word 'class' was historically used synonymously with 'set', before the modern distinction was made.
Reader's Guide
The notion of class is foundational to modern set theory and its applications. In Zermelo–Fraenkel set theory, classes are informal and must be reduced to formulas without class symbols. In von Neumann–Bernays–Gödel set theory, classes are axiomatized as basic objects, with sets defined as classes that are members of other classes. Morse–Kelley set theory goes further by allowing quantification over all proper classes in its class existence axioms, making it strictly stronger than NBG and ZFC. Other set theories, such as New Foundations, also give rise to proper classes because they do not postulate that all subclasses of a set are themselves sets. In category theory, a category whose collection of objects or morphisms forms a proper class is called a large category. The concept of class function generalizes the notion of function to classes, using a formula rather than a set. Overall, classes provide a rigorous way to discuss large collections while avoiding the paradoxes that plagued naive set theory.
Did You Know?
- A class that is not a set is called a proper class; a class that is a set is sometimes called a small class.
- In Quine's set-theoretical writing, the phrase 'ultimate class' is used instead of 'proper class'.
- The surreal numbers are a proper class of objects that have the properties of a field.
- In ZF set theory, classes have no formal status and formulas with classes must be reduced syntactically to formulas without classes.
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