Zermelo-Fraenkel set theory

Zermelo-Fraenkel set theory (ZF) is the standard foundation of modern mathematics: a formal system of axioms that defines what sets are, how they can be constructed, and what operations upon them are legitimate. Every mainstream mathematical object — numbers, functions, spaces, graphs — can be encoded as a set within ZF, and virtually every theorem in contemporary mathematics can be proved from the ZF axioms together with the Axiom of Choice (producing ZFC). Yet ZF is not merely a bookkeeping system for mathematicians. It is a description of an iterative universe of sets built by transfinite recursion from the empty set, and the structure of that universe determines what mathematical objects exist, what questions are decidable, and what principles of reasoning are valid.

The Axioms

The ZF axioms were formulated by Ernst Zermelo in 1908 and refined by Abraham Fraenkel and Thoralf Skolem in the 1920s. They replace the naive comprehension principle — for