Zermelo-Fraenkel set theory (ZF) is a foundational system for mathematics that describes how sets, collections of objects, can be constructed and manipulated. It includes several axioms that govern the behavior of sets, such as the existence of empty sets and the ability to form unions and power sets. The Axiom of Choice (AC) is an additional principle that states for any collection of non-empty sets, it is possible to select one element from each set. Together, ZF with AC, often referred to as ZFC, provides a robust framework for much of modern mathematics, allowing for the development of various mathematical concepts and structures.