Set theory
lastmod 2014-11-02created 2014-11-021 edits
Set theory (集合論, set theory) is the mathematical theory that studies sets, which are collections of abstract objects. Together with predicate logic, set theory is the foundation of most systems of the foundations of mathematics, and it logically underpins modern mathematics.
In naive set theory (naive set theory), a set is understood simply as a self-evident concept formed by collecting objects. The concept of sets covered in elementary and middle school curricula corresponds to this.
Axiomatic set theory, which emerged to resolve the contradictions of naive set theory, defines sets indirectly by specifying the axioms that sets and their membership relation satisfy. Here, sets and their membership relation can be seen as undefined terms, like points and lines in Euclidean geometry. In most cases, axiomatic set theory is not taught unless one majors in mathematics at university.