The Zermelo-Fraenkel set theory, often abbreviated as ZF, is a foundational system for mathematics that describes how sets can be constructed and manipulated. It was developed in the early 20th century by mathematicians Ernst Zermelo and Abraham Fraenkel. This theory provides a formal framework for understanding collections of objects, known as sets, and includes axioms that govern their behavior. One of the key features of Zermelo-Fraenkel set theory is its use of axioms, which are basic assumptions accepted without proof. These axioms help avoid paradoxes that can arise in naive set theory, such as the Russell's Paradox. When combined with the Axiom of Choice, it forms the Zermelo-Fraenkel set theory with Choice (ZFC), a widely accepted foundation for modern mathematics.