naive set theory

Naive set theory, developed by Georg Cantor, views a set as a collection of objects, where each object is either a member of the set or not. Sets are defined by their members, not by how they are described. For finite sets, elements are counted using positive integers, and the number of elements is the set's cardinality. Infinite sets are those that cannot be put into a one-to-one correspondence with a finite set of integers.1

A key principle is abstraction, which asserts the existence of a set for any given property or formula {x | S(x)}. However, an unrestricted version of this principle led to paradoxes, such as Russell's paradox, which demonstrated inconsistencies in early set theory. Modern axiomatic set theory, like Zermelo-Fraenkel (ZF), restricts this principle to avoid such paradoxes. Set theory also defines operations like union and intersection, and concepts like subsets and equivalence.12345