Zermelo-Fraenkel set theory (ZF)

Zermelo-Fraenkel set theory (ZF) is a foundational framework for mathematics that describes how collections of objects, called sets, work. It provides precise rules—axioms—about how sets can be created, combined, and related, ensuring consistency and avoiding paradoxes. In ZF, everything in mathematics is built from sets, including numbers, functions, and geometric concepts. It emphasizes principles like pairing sets, forming subsets, and the idea that sets are objects that can contain other sets, thus creating a rigorous basis for understanding and formalizing mathematical ideas.