Class (set theory)

In mathematics, a class is a group of mathematical objects, like sets, that all share a common property. Classes help us organize and talk about big collections of things without running into problems, especially one called Russell's paradox.

Some classes are also sets, and these are called small classes. But there are classes that are too big to be sets, known as proper classes. For example, the class of all ordinal numbers or the class of all sets is a proper class in many systems.

The way we define classes can change depending on the rules, or "axioms," we use in set theory. Some systems treat classes in a more informal way, while others, like von Neumann–Bernays–Gödel set theory, have strict rules about proper classes. In the work of Quine, these very large classes are sometimes called "ultimate classes" because they can't be part of any larger group.

Examples

In mathematics, a class is a group of objects that share a common property. For example, the collection of all groups, vector spaces, and many other algebraic structures are usually classes. These collections are so large that they are called "proper classes."

The surreal numbers are a proper class that behave like numbers. In set theory, many large collections, such as all sets, all ordinal numbers, and all cardinal numbers, are also proper classes. One way to show a class is proper is to match it with the class of all ordinal numbers.

Paradoxes

In set theory, some problems happen when we assume that every group of mathematical objects is a set. These problems help us understand that some groups, called classes, are too big to be sets.

For example, Russell’s paradox shows that the group of all sets that do not contain themselves cannot be a set. Similarly, the Burali-Forti paradox shows that the group of all ordinal numbers is also too big to be a set. These paradoxes teach us important lessons about how to organize mathematical objects safely.

Classes in formal set theories

ZF set theory does not include classes as official objects, so any idea involving classes must be rewritten using only sets. For example, instead of saying "A is the class of all sets x where x equals x," we say "for any set x, x is in A if and only if x equals x."

In other approaches, classes are treated as collections described by properties. For instance, the "class of all sets" can be thought of as all sets that satisfy the simple property "x equals x." Some set theories, like the von Neumann–Bernays–Gödel axioms, treat classes as the main objects and define sets as special kinds of classes. Other theories, such as Morse–Kelley set theory or New Foundations, also use classes in different ways, sometimes leading to stronger or more flexible systems.