Set

A set is one of the basic objects in the mathematical discipline of set theory, upon which most of apeirology is built. Although there's no technical way to define a set, a set is usually considered a collection of objects, and visualized as a bag. For example, the bag can be empty, yielding the empty set. In some systems of set theory, one has urelements, objects which aren't sets - however, a vast majority of set theory is pure set theory where everything eventually reduces to a set. For example, the Von Neumann ordinal assignment defines the natural numbers as nested bags, with zero being empty, and taking successor being adding the number to its own bag - that is, \(0 = \emptyset\) and \(a+1 = a \cup \{a\}\).

The discipline of set theory has developed various operations for constructing and comparing sets as well as rules for how sets are behaved, and what properties abstract, infinite sets such as ordinals have.