A set is a collection of distinct objects, called elements, with a definite membership criterion: for any candidate object, either it belongs to the set or it does not. Sets are specified by listing their elements ({1, 2, 3}) or by a property ({x : x is even}). Two sets are equal if and only if they have the same elements — a set is determined entirely by its membership, not by how it is described.
The axioms of set theory (Zermelo-Fraenkel with Choice, ZFC) regulate which collections count as sets, avoiding paradoxes like Russell’s (the set of all sets not containing themselves). The basic operations — union, intersection, complement, power set, Cartesian product — build new sets from existing ones.
Sets are the standard foundation for mathematics: functions are sets of ordered pairs, relations are sets of ordered pairs, numbers are constructed from sets. A category of sets (Set) has sets as objects and functions as morphisms, and many categorical constructions generalize set-theoretic ones.
In the semiotic universe, the elements of the Heyting algebra H are not sets but semantic values. However, the subsets of H (ordered by inclusion) form a Boolean algebra, and the subobject classifier Ω in a topos generalizes the role that {true, false} plays in Set — classifying which elements belong to a given subobject.