Most of modern mathematics rests on a handful of rules about sets
When paradoxes cracked the early theory of sets, mathematicians rebuilt it from a short list of axioms. The result, Zermelo-Fraenkel set theory with the axiom of choice, is now the usual foundation for nearly everything mathematicians prove, even though, by Gödel's work, it can never certify its own consistency.
Georg Cantor and Richard Dedekind launched the modern study of sets in the 1870s, treating any well-defined collection as a set. That freedom proved dangerous. Russell's paradox considers the collection of all sets that do not contain themselves and asks whether it contains itself; either answer contradicts itself. Mathematicians needed rules that would allow useful sets while forbidding troublesome ones.
Ernst Zermelo offered the first axiomatic system in 1908. In a 1921 letter Abraham Fraenkel pointed out that it could not prove the existence of certain large sets most mathematicians took for granted, and one axiom relied on a vague notion of a definite property. In 1922 Fraenkel and Thoralf Skolem independently fixed both problems, defining definite properties through first-order logic and adding the axiom schema of replacement. With John von Neumann's axiom of regularity, which rules out any set being a member of itself, the system took its modern form.
Its language is spare: one relation, membership, and nothing but pure sets, with no outside objects called urelements. Collections that are too big, like a set of all sets, simply do not exist in the theory, and new subsets can only be carved out of sets already known to exist. That restriction is precisely what blocks Russell's paradox. An extension called NBG lets mathematicians talk about such oversized collections, known as proper classes, more directly.
The initials tell the story of a long argument. ZF is the core system, and ZFC adds the axiom of choice, once hotly disputed and now standard. Later landmark results showed that choice is independent of the other axioms, as is Cantor's continuum hypothesis from ZFC itself, meaning neither can be proved or refuted from the rest.
Source: Zermelo–Fraenkel set theory
