ZFC

ZFC, short for Zermelo-Fraenkel set theory with the axiom of choice, is the standard foundational framework for much of mathematics. It formalizes sets and their membership relations in a single-sorted first-order language and aims to provide a rigorous basis for constructing mathematical objects while avoiding paradoxes through restricted set formation.

ZFC is defined by a collection of axioms. Extensionality asserts that two sets with the same elements are equal. The Axiom of Foundation (Regularity) prevents infinitely descending chains of membership. Pairing and Union guarantee the existence of pair sets and the union of a set. Power Set ensures that for any set there is a set of all its subsets. Infinity asserts the existence of an infinite set, typically modeled by the natural numbers. Replacement allows the image of a set under any definable function to be a set. Separation (also called the subset axiom) guarantees that for any property, a definable subset of any given set exists. Finally, the Axiom of Choice posits that from any collection of nonempty sets one can select a single element from each set.

Historically, Zermelo introduced an initial form of set theory, later refined by Fraenkel and Mostowski into ZF. The axiom of choice was added to form ZFC. The independence of the Axiom of Choice from ZF was shown by Gödel and Cohen: neither AC nor its negation can be proved from ZF, assuming ZF is consistent.

Today ZFC serves as the default foundation for most of ordinary mathematics, enabling formalization of most theorems. Some areas explore alternative foundations, such as ZF without AC, type theories, or category-theoretic foundations, but ZFC remains the most widely used framework.