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 big collections of things.

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 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."

Examples

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

The surreal numbers are a proper class that act like numbers. In set theory, many big 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 occur when we think that every group of mathematical objects can be a set. These problems show us that some groups, called classes, are too large to be sets.

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

Classes in formal set theories

ZF set theory does not use classes as real objects. So, when we talk about classes, we change the words to only use 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 ways, classes are groups we describe with rules. For example, the "class of all sets" is all sets that follow the rule "x equals x." Some set theories, like the von Neumann–Bernays–Gödel axioms, see classes as the main things and sets as special classes. Other theories, such as Morse–Kelley set theory or New Foundations, also use classes in different ways. This can make the systems stronger or more flexible.