Equivalents Applications and Consistency of the Axiom of Choice

Given any family of non-empty sets the Axiom of Choice (AC) gives us the power to choose one element from each member from the family. AC appears and is used in most modern mathematics and sciences more often than not implicitly. For example when we say \choose coset representatives we are indeed using AC. Upon its formal formulation after unconscious uses AC had been receiving heavy criticisms for years due to its non-constructive nature. However AC indeed gives us a richer mathematical (and scientific) world. We first provide as motivation some equivalents of AC and results requiring AC from various branches of mathematics. Then we prove the consistency of AC in ZF. Some results from AC e.g. the Banach{Tarski paradox are discussed.