Class (Set Theory)

Definition

In set theory, a class is a collection of sets defined by a property, which need not itself be a set. Formally, given a predicate $\varphi$, the class of all sets satisfying $\varphi$ is written $\{x\mid \varphi (x)\}$.

Every set is a class, but not every class is a set. A class that is not a set is called a proper class.

Motivation

The need for classes arises from Russell’s Paradox. Unrestricted comprehension — forming a set from any predicate — leads to contradiction. Restricting comprehension to sets (as in ZFC) resolves this, but leaves certain large collections, such as the collection of all sets, without a home. Classes provide a name for these collections without admitting them as sets.

Examples

• The von Neumann universe $V$: the class of all sets.
• The class of all ordinals $\mathrm{Ord}$.
• The class of all cardinals.
• The class of all groups (in the sense of set-theoretic algebra).

Each of these is a proper class: assuming any of them were a set leads to a contradiction.

Classes in Formal Set Theory

In ZF, classes are not formal objects — they are a convenient shorthand for predicates ($\mathrm{Set}\to \mathrm{Prop}$) . The statement $x\in \{y\mid \varphi (y)\}$ is simply an abbreviation for $\varphi (x)$.

In von Neumann–Bernays–Gödel (NBG) set theory, classes are first-class objects. Every set is a class, but only sets may be elements of other classes. NBG is a conservative extension of ZFC: every theorem of ZFC about sets is a theorem of NBG.

Relationship to Type Theory

In type theory, an analogous issue arises with universes. A universe $\mathcal{U}$ containing itself ($\mathcal{U}:\mathcal{U}$) leads to Girard’s Paradox, which is the type-theoretic analogue of Russell’s paradox. Universe hierarchies ${\mathcal{U}}_0:{\mathcal{U}}_1:{\mathcal{U}}_2:\cdots$ serve a role analogous to the set/class distinction.

Related Concepts

• Set
• Set Theory
• Russell’s Paradox
• Ordinal
• Tarski Universe
• Comprehension