The Zermelo-Fraenkel axioms, often abbreviated as ZF, are a set of foundational principles for set theory, which is a branch of mathematical logic. These axioms provide a framework for understanding how sets, or collections of objects, behave and interact. They help mathematicians avoid paradoxes and inconsistencies that can arise when dealing with infinite sets. One of the key features of the Zermelo-Fraenkel axioms is the Axiom of Choice, which is sometimes added to form ZFC. This axiom allows for the selection of elements from sets, even when no specific rule for selection is provided. Together, these axioms form the basis for much of modern mathematics, enabling rigorous proofs and the development of various mathematical structures.