Set

set-theory

Definition

Set

A set is a collection of distinct objects, called its elements, regarded as a single object. Membership of an element $x$ in a set $A$ is written $x\in A$, and a set is determined completely by its elements:

$$A=B⟺\forall x(x\in A⟺x\in B).$$

An ordinary set ignores multiplicity: writing an element more than once does not change the set. For example, $\{a,a,b\}=\{a,b\}$.

This intuitive collection-as-one-object view is expressed by Cantor’s conception of a set.