ZermeloFraenkelin

Zermelo-Fraenkel set theory, commonly abbreviated as ZF (and ZFC when the axiom of choice is included), is a formal system intended to provide a rigorous foundation for most of mathematics. It is built on the language of first-order logic with a single binary relation, membership (∈). The theory aims to formalize how sets are constructed and related while avoiding paradoxes that emerged from naive set theory.

The core of ZF consists of several axioms. Extensionality states that sets are determined by their elements. The existence of an empty set is guaranteed. Pairing, Union, and Power Set axioms describe basic ways to create new sets. Infinity asserts the existence of an infinite set. The Axiom of Separation (a schema) allows formation of subsets by properties defined in a formula. Replacement (a schema) handles images of sets under definable functions. Regularity (Foundation) forbids certain circular membership chains. These axioms collectively regulate how sets can be built and manipulated.

The axiom of choice (AC) is not a part of ZF but is added to form ZFC. AC has powerful consequences and is independent of the other axioms: neither provable nor refutable from ZF, assuming ZF is consistent. This independence extends to several mathematical propositions, including the generally known independence of the Continuum Hypothesis (CH) from ZFC, as established by Gödel and Cohen.

Zermelo-Fraenkel set theory is the standard framework for modern set theory and serves as a foundational backbone for much of mainstream mathematics, while alternative foundations (such as type theory or category theory) also exist.