Class (set theory)

https://en.wikipedia.org/wiki/Class_(set_theory) https://en.wikipedia.org/wiki/Subclass_(set_theory)

Informally, a class is a collection of sets that are unambiguously defined by a common property.

Sometimes, a proper class referes to a class whose elements are non-classes (a class that is not a set). And a small class (class that is a set) is used to refer to a class whose elements may be other classes.

Outside set theory, a class is sometimes a synonym for a set.

Subclass

A subclass is a class contained in some other class in the same way that a subset is a set contained in some other set.

That is, given classes A and B, A is a subclass of B if and only if every member of A is also a member of B. If A and B are sets, then of course A is also a subset of B. In fact, when using a definition of classes that requires them to be first-order definable, it's enough that B be a set; the axiom of specification essentially says that A must then also be a set.

As with subsets, the empty set is a subclass of every class, and any class is a subclass of itself. But additionally, every class is a subclass of the class of all sets. Accordingly, the subclass relation makes the collection of all classes into a Boolean lattice, which the subset relation does not do for the collection of all sets. Instead, the collection of all sets is an ideal in the collection of all classes. (Of course, the collection of all classes is something larger than even a class!)

A class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share.

The precise definition of "class" depends on foundational context. In ZF the notion of class is informal, whereas NBG axiomatizes the notion of "proper class" e.g. as entities that are not members of another entity.

In ZF, a class that is not a set 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.

A subclass is a class contained in some other class in the same way that a subset is a set contained in some other set.

That is, given classes A and B, A is a subclass of B iff every member of A is also a member of B. If A and B are sets, then of course A is also a subset of B. In fact, when using a definition of classes that requires them to be first-order definable, it's enough that B be a set; the axiom of specification essentially says that A must then also be a set.

As with subsets, the empty set is a subclass of every class, and any class is a subclass of itself. But additionally, every class is a subclass of the class of all sets.

Accordingly, the subclass relation makes the collection of all classes into a Boolean lattice, which the subset relation does not do for the collection of all sets. Instead, the collection of all sets is an ideal in the collection of all classes. Of course, the collection of all classes is something larger than even a class!